Overhead line Methodology

A methodology to design an overhead line

September 21, 2022

Soukayna Jermouni
Álvaro Benito Oliva
Ignacio Álvarez Iberlucea
Meyer Montagner Murcian 1

Félix Pérez Cicala

Álvaro Pajares Barroso

1
Provided essential software knowledge
 Abstract

Abstract

This methodology describes the design process to calculate an overhead line that connects the
so- lar plant’s station facility with the grid’s point of interconnection. The objective of this
document is to present the main steps followed to calculate the electrical and mechanical
characteristics of an overhead line.

An overhead line design goes through several stages from planning to execution. This document
will focus on explaining the stages that are in between. Following are the topics that will be
covered in this document to explain the design of the overhead line:

• The line’s blocks definition

• Selection of the line’s phase conductor
• Insulation coordination and clearances
• Selection of the line’s insulator
• Towers spotting
• Catenary and sags calculation
• Spans calculation
• Towers top-geometry calculation and selection of standard towers
• Earth wire calculation
• Electrical parameters calculation
• Tower forces calculation

Note: All the calculations that are presented in this document are carried out according to the
IEC and EN standards.

Overhead line 1
 Conten

Contents

Abstract 1

1 Overhead line criteria 7

1.1 Transmission and distribution criterion . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Design criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Line feasibility criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Terrain data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Block definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.1 Deflection angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5.2 Spans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Type of towers...............................................................................................................10
1.7 Type of insulators..........................................................................................................10

2 Phase conductor 12
2.1 Conductor’s type...........................................................................................................12
2.1.1 Medium voltage conductors.............................................................................12
2.1.2 High voltage conductors...................................................................................13
2.2 Selection criteria............................................................................................................13
2.2.1 Maximum admissible current...........................................................................13
2.2.2 Minimum tensile strength.................................................................................16
2.2.3 Voltage gradient................................................................................................16
2.3 Bundle calculation.........................................................................................................17

3 Earth Wire 19
3.1 Earth wire type..............................................................................................................19
3.2 Earth wire selection.......................................................................................................20

4 Tower Selection 21
4.1 Towers types..................................................................................................................21
4.1.1 Medium voltage towers....................................................................................21
4.1.2 High voltage towers..........................................................................................22
4.2 Towers selection criteria................................................................................................23
4.3 Circuit selection.............................................................................................................23

5 Insulation coordination 25
5.1 General procedure.........................................................................................................25
5.2 Determination of withstand voltage..............................................................................26

Overhead line 2
 Conten

6 Insulators 27
6.1 Insulator’s types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.1.1 Medium voltage insulator . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.1.2 High voltage insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2 Selection criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2.1 Electrical criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2.2 Mechanical criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Clearances 32
7.1 Electrical clearances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.1.1 Phase to phase clearances . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.1.2 Phase to earth clearances . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.2 Mid-span clearances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.3 Safety distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8 Tower geometry 41
8.1 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8.2 Peak distance calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8.3 Standard towers top-geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9 Electrical calculation 45
9.1 Electrical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9.1.1 Distributed Parameters Model . . . . . . . . . . . . . . . . . . . . . . . . 46
9.2 Electrical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
9.2.1 Geometrical Mean Distance . . . . . . . . . . . . . . . . . . . . . . . . . 48
9.2.2 Geometrical Mean Radius . . . . . . . . . . . . . . . . . . . . . . . . . . 50
9.2.3 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
9.2.4 Inductance and Reactance . . . . . . . . . . . . . . . . . . . . . . . . . . 51
9.2.5 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
9.2.6 Capacitance and Susceptance . . . . . . . . . . . . . . . . . . . . . . . . 53
9.3 Voltage drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9.4 Power factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.5 Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.5.1 Joule effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.5.2 Corona losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10 Mechanical calculation 58
10.1 Conductor Loads hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
10.2 Loads calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
10.2.1 Weight loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
10.2.2 Wind loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
10.2.3 Ice loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
10.2.4 Total loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
10.3 Catenary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
10.3.1 Catenary curve calculation . . . . . . . . . . . . . . . . . . . . . . . . . 62
10.3.2 Horizontal tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
10.3.3 State change equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
10.4 Tower forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
10.4.1 Tower load hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
10.4.2 Vertical forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Overhead line 3
 Conten

10.4.3 Transversal forces.............................................................................................68

10.4.4 Longitudinal forces...........................................................................................69

11 Tower spotting 70
11.1 Considerations...............................................................................................................70
11.2 Spotting process............................................................................................................70
11.2.1 Distance check..................................................................................................71
11.2.2 Ground clearance validation.............................................................................71
11.2.3 Valid towers spotting........................................................................................73

Bibliography 74

A Selecting a phase conductor and an insulator 76

A.1 Selection of the phase conductor...................................................................................76
A.2 Selection of the insulator...............................................................................................77

B Electrical parameters calculation 79

B.1 Resistance......................................................................................................................79
B.2 Inductance and reactance...............................................................................................79
B.3 Impedance......................................................................................................................80
B.4 Capacitance and susceptance.........................................................................................80
B.5 Voltage drop..................................................................................................................80
B.6 Joule losses....................................................................................................................81
B.7 Corona losses.................................................................................................................81

C Mechanical calculations 83
C.1 Calculation of the loads.................................................................................................83
C.2 Maximum horizontal tension.........................................................................................84
C.3 Horizontal tension calculation.......................................................................................86
C.4 Catenary calculation......................................................................................................86
C.5 Tower forces..................................................................................................................87
C.5.1 Vertical forces...................................................................................................88
C.5.2 Transversal forces.............................................................................................88
C.5.3 Longitudinal forces...........................................................................................89

Overhead line 4
 List of

List of Figures

1.1 The loadability factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Blocks definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Spans of the line.............................................................................................................10

4.1 MV line towers shape: (a)Single fork tower | (b)MV Double fork tower.....................22
4.2 HV line towers shape: (a) Single S tower | (b) Single Pi tower | (c) Double T tower
| (d) HV Double fork tower............................................................................................23

7.1 The swing angle.............................................................................................................36

7.2 Safety distances of the line............................................................................................39

8.1 Top-geometry of an S tower..........................................................................................42

9.1 Equivalent circuit of the short line model.....................................................................45

9.2 Equivalent circuit of the pi model.................................................................................46
9.3 Equivalent circuit of the distributed parameters model................................................46
9.4 Phases of a tower with one circuit for the GMD calculation........................................48
9.5 Phases of a tower with two circuits for the GMD calculation......................................49
9.6 Four conductors bundle.................................................................................................50

10.1 Catenary of a conductor................................................................................................63

10.2 Transversal forces of a deflected tower with wind.......................................................68

11.1 The distance check: (a) User line path | (b) Path filtering | (c) Possible towers
location | (d) Suspension towers are added
71
11.2 The ground clearance validation...................................................................................72
11.3 Tower spotting sample..................................................................................................73

Overhead line 5
 List of

List of Tables

2.1 The target allowable span lengths [Source: own elaboration]......................................16

2.2 The minimum tensile strength [Source: own elaboration]............................................16
2.3 Number of sub-conductors calculation [Source: Own elaboration]..............................18

3.1 The OPGW wire. Source ZTT catalog [6]....................................................................19

4.1 Tower types selection....................................................................................................23

4.2 Number of circuits calculation. [Source: Own elaboration].........................................24

6.1 Long-rod insulators according to IEC. Source IEC 60433 [14] and [15].....................28
6.2 Cap-and-pin insulators according to IEC. Source: [13] and [15]..................................29
6.3 The expected tensile strength for insulators..................................................................31

7.1 The minimum standard clearances pp and pe. Source: EN standard [17]....................33
7.2 The minimum standard clearances pp and pe [Source EN]..........................................38

8.1 Tower type Atorinillada N shape. [Source: Imedexsa [19]].........................................43

8.2 Tower type Condra S shape. [Source: Imedexsa [19]].................................................43
8.3 Tower representing the Pi tower. [Source: Imedexsa [19]]..........................................43
8.4 Tower type Condor N Doble. [Source: Imedexsa [19]]................................................43
8.5 Tower type Icaro N shape. [Source: Imedexsa [19]]....................................................44

10.1 Ice loads for different countries for a conductor with a 30 mm diameter. [Source:
[3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

11.1 The maximum tower height. Source: Imedexsa [19] . . . . . . . . . . . . . . . . 72

C.1 Mechanical data of the conductor "160-A1/S2A" . . . . . . . . . . . . . . . . . . 83

C.2 Sags and tensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
C.3 Inputs for tower forces calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Overhead line 6
 Chapter 1. Overhead line

Chapter 1

Overhead line criteria

In this chapter the overhead line’s main elements and design criteria will be described.

1.1 Transmission and distribution criterion

The overhead line that is designed in this document can be of a transmission or of a distribution
nature depending on the voltage and the type of the plant’s interconnection facility. When a
user selects a switching and breaking station, the voltage will be of a medium level as the station
does not perform a voltage step-up function; hence, the power will be directly evacuated in
a distribution line to the utility’s distribution substation.

On the other hand, in the case of choosing a substation as the plant’s interconnection
facility, the station will have a high voltage at the receiving end which will define a
transmission line to evacuate the generated capacity.

Furthermore, the minimum medium voltage considered is 5kV and the maximum is 45kV.
Mean- while, the high voltage level starts from 46kV up to 400kV.

1.2 Design criteria

In this methodology, several electrical requirements in designing an overhead line were consid-
ered, these considerations are defined to guarantee the stability and the capability of the line.

The overhead line is designed to connect the plant’s interconnection facility at its voltage level;
hence, the maximum high voltage accepted is 400kV as it is the highest acceptable voltage in
the plant’s substation.

Furthermore, to ensure the capability of the line, a maximum capacity that the line can evacuate
is limited by both its voltage and length. Therefore, a thermal limit as well as a voltage drop and
a power loss limit must be respected. In this regard, a voltage drop and a power loss of 5% [1] is
considered as the maximum permissible.

On the other hand, the maximum voltage gradient on the conductor surface that is considered
in selecting the conductor section is designed as 17kV/cm [1].

Overhead line 7
 Chapter 1. Overhead line

Regarding the reactive power, no way of compensation is considered in this methodology. The
power factor at the receiving end is calculated accordingly and its calculation will be presented
through this methodology.

1.3 Line feasibility criterion

The design of the line conforms to some criteria that define the possible voltages, capacities,
and line lengths that comply with the voltage drop and power loss limits to perform efficiently.
In this methodology, as mentioned in the previous section, a limit of 5% will be considered as
the maximum permissible.

Based on several test cases that include different line lengths (short, medium, and long),
different voltage levels from 5kV to 400kV, and different capacities, the resulting line
feasibility is con- strained to comply with the voltage drop and power loss limitation. This is
also called electrical loadability of the line. In Figure 1.1 the maximum line lengths depending
on the capacity and voltage that will guarantee the performance of the line are presented,
these lengths are results of several tests as mentioned.

Figure 1.1: The loadability factor

1.4 Terrain data

The elevation data under the overhead line is calculated to perform the spotting of the
towers within the line. An inverse distance weighted (IDW) interpolation of the elevation data
is applied when necessary and it considers the coverage of the digital elevation model.
Wherever there is a gap in the elevation, the interpolator will fill the gap accordingly. The
algorithm used by the interpolator is explained thoroughly in [2].

1.5 Block definition

The overhead line is defined as blocks of towers that are grouped based on the deflection of the
path, an illustration of blocks and deflections are shown in Figure 1.2 Consequently, the first
and last towers of each block are tension towers. Moreover, every two adjacent blocks will
share one tower.

Overhead line 8
 Chapter 1. Overhead line

Figure 1.2: Blocks definition

1.5.1 Deflection angle

The deflection angle is the angle by which the line deviates and by which a block is defined. For
a new block to be considered in the line, the deflection angle should be higher than 2° or lower
than -2°. To calculate the deflection angle a simple geometric function is used to get the angle
between the three towers, and from it we deduce the deflection angle of the line.

1.5.2 Spans
The line’s blocks are themselves characterized by the different spans that are used further in
mechanical calculations. In this section, the types of the spans considered in the calculation are
depicted.

Length span
The length span is the distance between two adjacent towers taking into account the elevation
difference between the two towers.

Weight span
The weight span is the distance between the lowest points of the sags in two adjacent spans. The
lowest points are calculated from the catenary of the line. Moreover, the weight span represents
the vertical load applied to the tower due to the weight of the conductor.

Wind span
The wind span is defined as half of the sum of two adjacent spans in a profile view. The wind
span represents the horizontal load applied on the tower due to the wind blowing on the
conductor.

In Figure 1.3 the different spans of the line are illustrated.

Ruling span
On the other hand, the ruling span is considered as a mean span within a block. Each block
has its ruling span that is used to approximately represent its mechanical characteristics. The

Overhead line 9
 Chapter 1. Overhead line

Figure 1.3: Spans of the line

theoretical ruling span is calculated to simplify the sagging calculation of the conductors, and it
is calculated using Equation (1.1) by making the assumption that the towers are of equal
elevations and that the horizontal tension is constant throughout the block.

√︂
𝑆𝑟 = (1.1)
Σ𝑆
Σ𝑆3
Where:

• 𝑆𝑟 is the theoretical ruling span [m]

• 𝑆 is a span length [m]

1.6 Type of towers

The towers of an overhead line can be of different types depending on the function of the line,
the number of circuits, the utility’s constraints as well as the country’s requirements. They
determine an important part of the line’s investment as well as the reliability of the line.

The towers that are used throughout this methodology are lattice steel towers of three types:

• Suspension tower: It carries the conductor when the line is straight, the longer the blocks
with suspension towers the better the investment of the line would be.
• Angle tower: It carries the conductor where the line deflects and changes direction.
Unlike the suspension towers, they support more tensile forces from the conductors.
• Dead-end towers: They are the towers at both ends of the line, they are usually connected
to the substation’s portals that generally leads to higher load at both ends.

1.7 Type of insulators

The insulators are another important component in the overhead line, they are installed to con-
nect the live conductors with the earthed towers. Their type is defined based on the type of the
tower they are connected to. In this methodology two types of insulators are considered:

Overhead line 1
 Chapter 1. Overhead line

• Suspension insulator: It is the insulator connecting the conductor to the suspension tower
when the line is straight.
• Tension insulator: It is the insulator that attaches the conductors to angle or dead-end
towers, it always follows the straining of the conductor.

Overhead line 1
 Chapter 2. Phase

Chapter 2

Phase conductor

The overhead line conductors correspond to up to 50% of the total investment of the line [ 3];
hence, the importance of the selection criteria of the conductor’s section and its bundles. In this
chapter, the type of the phase conductors that are considered will be described, the selection
consideration will be depicted, and the bundle calculation will be presented.

2.1 Conductor’s type

The overhead line conductors are usually bare conductors not insulated and stranded that can
be made of one material or a composition of different materials.

The conductors that were chosen in this methodology are of two types, conductors that are used
for medium voltage lines and made of Aluminum and conductors that are used in high voltage
lines which are made of a composition of Aluminum and steel. Both types of conductors are
designed following the IEC standard 61089 [3].

2.1.1 Medium voltage conductors

For the distribution lines, AAAC conductors are used which are made solely of Aluminum alloy
and are characterized by their lighter weights and lower electrical losses. The designation of an
AAAC conductors according to IEC [3] is identified by a number that corresponds to the section
of the conductor and a code that refers to the material used.

From the standardized conductors, several parameters are extracted such as the cross-section,
the number of stranding, the nominal diameter, the weight, and the rated strength of the
conductor.

Moreover, other parameters are considered based on [3] to calculate the resistance of the con-
ductor as well as its mechanical characteristics; these parameters are:
−8
The resistivity of an Aluminum conductor of type A and B being: 𝜌 = 3.27 · 10 Ω𝑚
−6◦
The coefficient of linear thermal expansion of an Aluminum alloy conductor being 𝛼 = 23 · 10 𝐶 −1

And the elasticity of the Aluminum alloy conductor being 𝐸 = 55.6 · 109𝑃𝑎

Overhead line 1
 Chapter 2. Phase

2.1.2 High voltage conductors

As for the transmission line, the ACSR conductors are chosen for the high voltage. The ACSR
conductors are bare and made of layers of Aluminum wires on a galvanized steel wires on the
core. This conductor is characterized by a high tensile strength and suitability in large spans.
The ACSR conductors used in this methodology are all designed according to IEC [3].

This latter’s designation is defined with a number that corresponds to the cross-section of the
conductor and two codes that represent the composition materials. Using the standardized con-
ductors, some mechanical parameters are extracted such as the section, the number of stranding,
the nominal diameter, the unitary weight, and the rated strength of the conductor.

These previous parameters along with others that are specific to ACSR conductors are used to
select the right conductor of the line and to calculate further mechanical characteristics. The
specific parameters are:

The elasticity of the Aluminum steel reinforced of the conductor being 𝐸 = 190 · 109𝑃𝑎

The coefficient of linear thermal expansion of an Aluminum steel reinforced conductor being
− ◦
𝛼 = 11.5 · 10 6 𝐶 −1

2.2 Selection criteria

The selection of a distribution or a transmission conductor relies mainly on the current loading;
generally, the operating current leads the conductor to run at a higher temperature,
neverthe- less, the conductor’s temperature should not exceed an admissible limit in order to
maintain its mechanical strength.

Therefore, two main criteria are investigated, an electrical criterion to check the thermal limit of
the conductor and a mechanical criterion that guarantee the strength withstand of this latter.

In addition, in this methodology, the conductors undergo a selection based on the voltage gra-
dient, the voltage drop and the power loss limits. The compliant conductor, therefore, must
withstand the thermal unit, comply with the voltage gradient and voltage drop maximum limits,
and must withstand the corresponding tensile strength in the line.

2.2.1 Maximum admissible current

For an overhead line to be reliable, it must be able to withstand the electrical load defined by the
operating current. Hence, the conductor’s cross-section design must conform with the maximum
admissible current of the line.

For given ambient conditions, the maximum admissible current is calculated based on the con-
ductor’s predetermined temperature and its DC resistance at maximum temperature [4]. To cal-
culate the DC resistance of the conductor at maximum temperature Equation (2.1) is used:

𝑅𝑇 = 𝑅𝑇𝑓 [1 + 𝛼 (𝑇𝑚𝑎𝑥 − 20)] (2.1)

Where:

Overhead line 1
 Chapter 2. Phase

• 𝑅𝑇 is the electrical resistance of the conductor per unit length at maximum temperature T
[Ω/𝑚]
◦
• 𝑅𝑇𝑓 is the DC resistance per unit length at a reference temperature being 20 𝐶 [Ω/𝑚]
−3 ◦
• 𝛼 is the variation of the resistance with temperature being 4.03 · 10 [1/ 𝐶]
◦
• 𝑇𝑚𝑎𝑥 is the maximum temperature being 80 [ 𝐶]

𝜌
𝑅𝑇𝑓 = (2.2)
𝑠
Where:

• 𝑅𝑇𝑓 is the electrical resistance of the conductor per unit length at reference temperature T
◦
being 20 𝐶 [Ω/𝑚]
• 𝜌 is the resistivity of the conductor [Ω · 𝑚]
• 𝑠 is the cross-section of the conductor [𝑚2]

To calculate the maximum admissible current a heat balance at the conductor should be reached;
this heat balance is determined based on the solar heat gained by the conductor surface, the heat
loss by convection, and the heat loss caused by radiation of the conductor.

Using the unit length DC resistance at maximum temperature of the conducto;, the heat
balanced is reflected in Equation (2.3):

√
𝐼𝑚𝑎𝑥 = ︂ 𝑁𝑅 + 𝑁𝐶 − 𝑁𝑆 (2.3)
𝑅𝑇

Where:

• 𝐼𝑚𝑎𝑥 is the maximum admissible current for one conductor [𝐴]

• 𝑁𝑅 is the heat loss by radiation of the conductor [𝑊 /𝑚]
• 𝑁𝐶 is the heat loss by convection [𝑊 /𝑚]
• 𝑁𝑆 is the solar heat gain by the conductor’s surface [𝑊 /𝑚]
• 𝑅𝑇 is the electrical resistance of the conductor per unit length at reference temperature T
◦
being 20 𝐶 [Ω/𝑚]

- Heat loss by radiation:

Given the nominal diameter of the conductor, the heat loss by radiation is given by the following
equation:

𝑁𝑅 = 𝑘 · 𝜋 · 𝑑 · 𝐾𝑒 (𝑇𝑚𝑎𝑥 4 − 𝑇𝑎𝑚4) (2.4)

Where:

• 𝑁𝑅 is the heat loss by radiation of the conductor [𝑊 /𝑚]

−8
• 𝑘 is the Stefan-Boltzmann constant being 5.67 · 10 [𝑊𝑚−2𝐾 −4]

Overhead line 1
 Chapter 2. Phase

• 𝑑 is the nominal diameter of the conductor [𝑚]

• 𝐾𝑒 is the emissivity coefficient with respect to a black body
• 𝑇𝑚𝑎𝑥 is the maximum temperature being 353 [𝐾 ]
• 𝑇𝑎𝑚 is the ambient temperature being 318 [𝐾 ]

- Heat loss by convection:

The heat loss by convection of the conductor is calculated as the following:

𝑁𝐶 = 𝜆 · 𝑁𝑢 · 𝜋 · (𝑇𝑚𝑎𝑥 − 𝑇𝑎𝑚) (2.5)

Where:

• 𝑁𝐶 is the heat loss by convection [𝑊 /𝑚]

• 𝜆 is the thermal conductivity of the air in contact with the conductor [𝑊 /(𝐾 · 𝑚)]
• 𝑁𝑢 is the Nusselt number calculated in Equation (2.6)
• 𝑇𝑚𝑎𝑥 is the maximum temperature being 353 [𝐾 ]
• 𝑇𝑎𝑚 is the ambient temperature being 318 [𝐾 ]

The Nusselt number is calculated as:

𝑁𝑢 = 0.65 · 𝑅𝑒 0.2 + 0.23 · 𝑅𝑒 0.61 (2.6)

Where:

• 𝑁𝑢 is the Nusselt number

• 𝑅𝑒 is the Reynolds number calculated in [eq]

The Reynolds number is calculated following the equation:

−1.78
𝑅𝑒 = 1.644 · 109 · 𝑑 · 𝑣 · [𝑇𝑎𝑚 + 0.5 · (𝑇𝑚𝑎𝑥 − 𝑇𝑎𝑚)] (2.7)

Where:

• 𝑅𝑒 is the Reynolds number

• 𝑑 is the the nominal diameter of the conductor [𝑚]
• 𝑣 is the wind speed being 2.016 (0.5 m/s) [𝑘𝑚/ℎ]
• 𝑇𝑚𝑎𝑥 is the maximum temperature being 353 [𝐾 ]
• 𝑇𝑎𝑚 is the ambient temperature being 318 [𝐾 ]

- Solar heat gain:

The solar heat gain by the conductor’s surface is calculated by the following equation:

Overhead line 1
 Chapter 2. Phase

𝑁𝑆 = 𝑌 · 𝑑 · 𝑆𝑖 (2.8)

Where:

• 𝑁𝑆 is the solar heat gain by the conductor’s surface [𝑊 /𝑚]

• 𝑌 is the solar radiation absorption coefficient being 0.8
• 𝑑 is the nominal diameter of the conductor [𝑚]
• 𝑆𝑖 is the intensity of solar radiation being 1045 [𝑊 /𝑚2]

2.2.2 Minimum tensile strength

After selecting the conductors that withstand electrically, the mechanical criterion should be
con- sidered to guarantee a compliant conductor. Depending on the voltage of the line, a target
span is defined as seen in Table 2.1; if this latter is between a certain range, a corresponding
minimum tensile strength is selected for the compliant conductor to withstand as seen in Table
2.2.

Table 2.1: The target allowable span lengths [Source: own elaboration]
Rated voltage
Target span [m]
𝑈 [kV]
⩽24 100
⩽220
⩽132 250
200
⩽400 300
>400 400

The minimum tensile strength values depending on the target spans were obtained with an em-
pirical approach which consisted in analyzing different lines with different conductors under
several voltage levels. As the tensile strength is directly related to the catenary of the conduc-
tor, it is concluded that the weaker conductors will result in higher catenaries and to avoid not
respecting the target span lengths, a minimum tensile strength limit is defined based on the al-
lowable span length as seen in Table 2.2.

Table 2.2: The minimum tensile strength [Source: own elaboration]

Minimum Tensile Strength [N]
Target span [m]

⩽100 20000
⩽200 45000
⩽250 60000
⩽300 75000
>300 90000

2.2.3 Voltage gradient

The voltage gradient or the critical surface gradient is the electrical potential across the
conductor surface. When the electrical field reaches the critical surface gradient, Corona effect
occurs which causes extra losses. Hence, the voltage gradient should be considered when
selecting the cross- section of a conductor and should be limited; consequently, in this
methodology the voltage gradient is limited to 17kV/cm [5].

Overhead line 1
 Chapter 2. Phase

The voltage gradient of a conductor is calculated using the following Equation (A.7)

𝐸= �
2 𝐶𝑖 [1 + 2 · (𝑟 /𝑠) (𝑛2 − 1) · 𝑠𝑖𝑛(𝜋 /𝑛2)] √ (2.9)
� 𝜋𝜖0 · 𝑛2 · 𝑟 3 · 100

Where:

• 𝐸𝑖 is the voltage gradient [𝑘𝑉 /𝑐𝑚]

• 𝐶𝑖 is the capacitance per unit length of a conductor, the capacitance is calculated using
Equation (9.36) or Equation (9.35) [𝐹 /𝑚]
−12
• 𝜖0 is the dielectric constant being 8.854 · 10 [𝐹 /𝑚]
• 𝑛2 is the number of conductors per bundle
• 𝑟 is the radius of the sub-conductor [𝑚]
• 𝑠 is the distance between the sub-conductors being 400 [𝑚𝑚]
• 𝑈 is the rated voltage [𝑘𝑉 ]

In addition to the voltage gradient limit, the selected cross-section conductor must comply with
the voltage drop and losses limit which are considered as 5% based on [3].

2.3 Bundle calculation

The number of sub-conductors per phase are calculated based on the voltage and capacity of
the overhead line. Considering different possible arrangements depending on the voltage level,
the number of conductors per bundle is selected based on the capacity of the corresponding
arrangement.

Furthermore, in the case of not finding a compliant conductor per phase according to the voltage
gradient limit, the number of sub-conductors per bundle is increased till a convenient cross-
section is selected.

Overhead line 1
 Chapter 2. Phase

Table 2.3: Number of sub-conductors calculation [Source: Own elaboration]

Rated voltageCapacity
Bundle number
𝑈 [kV] 𝑆 [MVA]
⩽15 ⩽10 1
>20
⩽20 42
⩽30 ⩽50 1
⩽100 2
>100 4
⩽66 ⩽160 1
⩽320 2
>320 4
⩽132 ⩽160 1
⩽640 2
>640 4
⩽220 ⩽340 1
>1360
⩽1360 42
<400 ⩽2000 2
⩽2500 3
>2500 4

Overhead line 1
 Chapter 3. Earth

Chapter 3

Earth Wire

Lightning strike is one of the main reasons behind the sudden outages of an overhead line, and
the earth wire comes as a protection schema to reduce these unexpected outages.

Therefore, the earth wire’s main function is to not only to protect the phase conductors
from possible lightning but also to return the phase-to-earth short-circuit current. Consequently,
they should be designed and specified adequately to serve their function.
◦ ◦
In addition, an earth wire is installed with a shield angle that is defined between 10 and 35 ; in
◦
this methodology, the earth wires are designed with a shield angle of 30 .

3.1 Earth wire type

The earth wires can be of steel combined with low aluminum material to take advantage of their
high level of conductivity. However, recently the use of an optical ground wire has become
more relevant.

The earth wires chosen in this methodology are of optical fiber wire type (OPGW) to extend
their purposes and be used to carry telecommunication signals as well. The said earth wires are
obtained from [6] and designed according to IEC 60794-4 and IEC 61395.

The selected four OPGW wires are listed in table 3.1

Table 3.1: The OPGW wire. Source ZTT catalog [6]

Short-circuit capacity [𝑘𝐴2 · 𝑠]

Code OPGWDiameterNumberWeightMaximumSection [mm]of fibers[kg/m]load [N][mm2]
Resistance [Ω/𝑚]

1C 1/36B1 10.2 36 0.394 67800.0 0.000054 13.9 1.58

L-48B1-85 12.30 48 0.540 85600.0 0.000085 43.5 0.75
YS-2C 1/48B1 15.25 48 0.716 93800.0 0.000133 138.1 0.33
2S 1/48B1 17.2 48 0.796 106300.0 0.000165 213.7 0.270

Overhead line 1
 Chapter 3. Earth

3.2 Earth wire selection

The earth wires chosen in this methodology are of different sizes, to select the adequate size for
a certain line, an electrical criteria based on the short-circuit current is considered.

Based on the line’s maximum voltage, the design short-circuit current is calculated according to
the IEC standard [7], [8], [9], and [10] moreover, the short-circuit time is taken as 0.3s. Hence,
the short-circuit design capacity is calculated using Equation (3.1) and compared to the earth
wire’s short-circuit capacity. The earth wire that withstands the said capacity is selected.

𝑆𝑠𝑐 = (𝐼𝑠𝑐 /1000) 2 · 𝑇𝑘 (3.1)

Where:

• 𝑆𝑠𝑐 is the short-circuit capacity [𝑘𝐴2 · 𝑠]

• 𝐼𝑠𝑐 is the design short-circuit current [𝑘𝐴]
• 𝑇𝑘 is the short-circuit duration set as 0.3 [𝑠]

Overhead line 2
 Chapter 4. Tower

Chapter 4

Tower Selection

The overhead line towers are another important component in granting the right-of-way as well
as in the investment of the power line, depending on their design and material, the cost of the
line can change remarkably. Therefore, the towers not only define the aesthetic of the line but
also determine its reliability by withstanding the conductor’s forces and loads.

In this chapter the overhead line’s towers will be described, their types upon voltage level and
number of circuit selection criteria will be presented.

4.1 Towers types

The overhead line towers shape differ from country to country and from a utility operator to
another. However, in this methodology, there are six different tower shapes designed for both
medium and high voltages, for simple and double circuits, and with one earth wire or two earth
wires. Furthermore, the towers material that is used in this methodology is of lattice steel for all
voltage levels.

4.1.1 Medium voltage towers

The chosen towers to be used for the MV lines are of two different shapes:

• Single fork tower: is a tower with one circuit (Simplex) and one earth wire arrangement
and it has the shape of a fork with its three crossarms being in one side.
• MV Double fork tower: is a tower with two circuits (Duplex) and two earth wires
ar- rangement and it has the shape of two forks with one circuit crossarms being in
one side and the other circuit in the opposite side.

The two MV towers are illustrated in Figure 4.1 following the order listed above.

Overhead line 2
 Chapter 4. Tower

Figure 4.1: MV line towers shape: (a)Single fork tower | (b)MV Double fork tower

4.1.2 High voltage towers

On the other hand, the four high voltage line towers are designed as follows:

• S shape tower: is a tower with one circuit (Simplex) one earth wire arrangement and
it has a shape of an S letter with 2 cross arms in one side and the third in the opposite
side.
• Single Pi tower: is a tower with one circuit (Simplex) and two earth wires
arrangement; it has the shape of the Greek letter Pi with the three phases aligned
horizontally.
• Double T tower: is a tower with two circuits (Duplex) and two earth wires
arrangement and it has the shape of the letter T with two levels, one circuit cross-arms
being in one side and the other circuit in the opposite side.
• HV Double fork tower: similar to the MV double fork tower, it is a tower with two
circuits (Duplex) and two earth wires arrangement and it has the shape of two forks
with one circuit cross-arms being in one side and the other circuit in the opposite side.
The difference with its MV similar is the middle cross-arms being longer than the others.

The HV towers shape are shown in the Figure 4.2 following the order listed above.

Overhead line 2
 Chapter 4. Tower

Figure 4.2: HV line towers shape: (a) Single S tower | (b) Single Pi tower | (c) Double T tower |
(d) HV Double fork tower

4.2 Towers selection criteria

Several aspects are considered in selecting a tower design, such as its impact in land use, the
ability to evacuate the necessary power, or the visual impact it will have on the landscape.

To select the right tower in this methodology, a voltage and circuit arrangement criteria were
considered. For each voltage level, MV or HV, depending on the number of circuits as well as
the voltage itself, a tower type is selected. The different voltage levels considered in the towers
selection is presented in Table 4.1.

Table 4.1: Tower types selection

Circuit arrangement
Voltage level [kV]
Voltage level Tower type

MV Simplex – Single fork tower

Duplex – MV Double fork
HV Simplex ⩽245 Single S tower
Simplex >245 Single Pi tower
Duplex ⩽245 HV double fork tower
Duplex >245 Double T tower
4.3 Circuit selection
The tower’s type selection is based on the number of circuits, that in its turn depends on the
voltage and capacity evacuated in the line. For a certain level of voltage, the circuit
arrangements is chosen to be simplex or duplex upon how much capacity is transmitted. In
the Table 4.2 , the possible arrangement used in the number of circuit selection, hence, the
tower type, are presented.

Overhead line 2
 Chapter 4. Tower

Table 4.2: Number of circuits calculation. [Source: Own elaboration]

Rated voltageCapacity
Number of circuits
𝑈 [kV] 𝑆 [MVA]
⩽15 ⩽4 1
⩽30 ⩽20
>4 12
>20 2
⩽66 ⩽80 1
>80 2
⩽132 ⩽320 1
⩽220 ⩽680
>320 12
>680 2
⩽400 ⩽1400 1
>1400 2

Overhead line 2
 Chapter 5. Insulation

Chapter 5

Insulation coordination

As mentioned in a previous chapter, the reliability of an overhead line is impacted by the elec-
trical and mechanical performance of the line. Consequently, tower electrical clearances play
an important role to reach a good mechanical performance. More particularly, the design of the
insulation coordination under, not only, temporary stresses but also different over-voltages is
crucial in defining the electrical clearances.

In this chapter, the insulation coordination design of the overhead line is described in obedi-
ence with the IEC standard [11]. The selection of the insulation levels will be explained and the
calculation of the withstand voltage to ensure the insulation of the system is stated.

5.1 General procedure

The insulation coordination procedure has an aim to guarantee a low probability of line damage
caused by over-voltages and it consists of treating several parameters to reach its objective.

The voltages are classified into two classes based on the IEC standard [11], and for each class
only some of the over-voltages are considered to calculate the withstand voltage. For class I
voltages (From 1kV to 245kV) both the temporary and the fast-front over-voltages are
considered. On the other hand, for class II voltages (Starting from 245kV) the fast-front and
slow-front over-voltages are considered in the calculation.

A detailed review of the procedure is shown in the substation methodology [12] chapter 4 where
the procedure details for class I and II are presented according to the IEC standard.

However, the insulation coordination procedure of an overhead line has few considerations that
are different from the procedure followed for the substation. To calculate the power frequency
over-voltage, the discharge factor of the earth fault for the overhead line calculation is
considered as 1.3 instead of 1.4 for the substation. Moreover, the defect factor is also considered
as 1.3 for the overhead line.

In addition to that, another difference in determining the withstand voltage for the overhead
line is the termination of the procedure followed in the substation when finding the required
withstand voltage [11].

Finally, normalized voltage levels are not considered in this calculation.

Overhead line 2
 Chapter 5. Insulation

5.2 Determination of withstand voltage

Calculating the required withstand voltage follows the same steps found in the substation
method- ology [12]. The power-frequency, slow-front and the fast-front over-voltages are
calculated based on the maximum voltage of the system by calculating both phase-phase and
phase-earth over- voltages considering the altitude of the project.

Furthermore, the statistical slow-front voltage is calculated to determine the clearances; it is

calculated based on the primary slow front over-voltage as follows:

√︁
𝑈𝑒𝑡 + 0.25
𝐸2 = · 𝑈 · 2/3 (5.1)
1.

Where:

• 𝐸2 is the statistical switching over-voltage [𝑝.𝑢]

• 𝑈𝑒𝑡 is the truncation value of the cumulative distribution of the phase-to-earth over-
voltages in p.u
• 𝑈 is the maximum voltage of the system [𝑘𝑉 ]

Overhead line 2
 Chapter 6.

Chapter 6

Insulators

The insulator plays important electrical and mechanical roles in the overhead line design, it
is, therefore, necessary to design it considering the insulation performance and the mechanical
withstand.

In this chapter the insulator’s types upon voltage level are described and the criteria used to
select the adequate insulator is presented.

6.1 Insulator’s types

The insulators of an overhead line come in different types and shapes with different materials;
in this methodology, the insulators are designed according to the IEC standard [13]. Namely,
long-rod and cap and pin insulators made of polymer and glass respectively are used to select
the correct suspension and tension insulators of the line.

6.1.1 Medium voltage insulator

For the medium voltage lines, the long-rod insulators are chosen for both suspension and angle
towers. The long-rod insulators are characterized by their high reliability especially under high
pollution conditions. This type of insulator comes with two different caps, in this methodology
only the clevis and tongue cap type is considered.

In the following Table 6.1, the insulators used in this methodology are listed with their corre-
sponding lightning impulse, minimum failing load, diameter and maximum length.

Overhead line 2
 Chapter 6.

Table 6.1: Long-rod insulators according to IEC. Source IEC 60433 [14] and [15]
lightningFailing impulse [V]load [N] MaximumMinimum length [m]creepage [m]
Designation Diameter [m]

L40 C170 170000 40000 0.16 0.4 0.576

L60 C170 170000 60000 0.16 0.42 0.576
L100 C170 170000 100000 0.18 0.475 0.576
L100 C250 250000 100000 0.18 0.6 0.832
L100 C325 325000 100000 0.18 0.9 1.16
L100 C450 450000 100000 0.18 1.12 1.968
L100 C550 550000 100000 0.18 1.27 1.968
L120 C325 325000 120000 0.2 0.905 1.16
L120 C450 450000 120000 0.2 1.12 1.968
L120 C550 550000 120000 0.2 1.275 1.968
L120 C650 650000 120000 0.2 1.465 2.32
L160 C325 325000 160000 0.21 0.92 1.16
L160 C450 450000 160000 0.21 1.135 1.968
L160 C550 550000 160000 0.21 1.29 1.968
L160 C650 650000 160000 0.21 1.465 2.32
L210 C325 325000 210000 0.22 0.94 1.16
L210 C450 450000 210000 0.22 1.155 1.968
L210 C550 550000 210000 0.22 1.31 1.968
L210 C650 650000 210000 0.22 1.5 2.32
L250 C550 550000 250000 0.23 1.335 1.968
L250 C650 650000 250000 0.23 1.53 2.32
L300 C550 550000 300000 0.24 1.365 1.968
L300 C650 650000 300000 0.24 1.56 2.320
L330 C550 550000 330000 0.25 1.4 1.968
L330 C650 650000 360000 0.24 1.595 2.32
L360 C550 550000 360000 0.25 1.41 1.968
L360 C650 650000 360000 0.25 1.6 2.32
L400 C550 550000 400000 0.26 1.46 1.968
L400 C650 650000 400000 0.26 1.66 2.32
L420 C550 550000 420000 0.26 1.46 1.968
L420 C650 650000 420000 0.26 1.66 2.32
L530 C550 550000 530000 0.27 1.52 1.968
L530 C650 650000 530000 0.27 1.72 2.32
L550 C550 550000 550000 0.27 1.52 1.968
L550 C650 650000 550000 0.27 1.72 2.32
CSC-750/2720 750000 210000 0.2 1.395 2.72
CSC-950/3920 950000 210000 0.25 1.775 3.92
CSC-1050/3920 1050000 210000 0.2 1.97 3.92

6.1.2 High voltage insulator

On the other hand, the cap-and-pin insulators were chosen to select the appropriate insulators
for the high voltage level. The cap-and-pin insulator is known for its high mechanical strength
and characterized by larger creepage distance. In Table 6.2 the cap-and-pin insulators used in
this methodology are listed.

Overhead line 2
 Chapter 6.

Table 6.2: Cap-and-pin insulators according to IEC. Source: [13] and [15]
lightningFailing impulse [V]load [N] Maximum length [m] Minimum creepage [m]
Designation Diameter [m] Weight [kg]

U40B 70000 40000 0.11 0.175 1.6 0.19

U70BS 100000 70000 0.127 0.255 3.6 0.295
U70BL 100000 70000 0.146 0.255 3.6 0.295
U100BS 100000 100000 0.127 0.255 3.9 0.295
U100BL 100000 100000 0.146 0.255 4 0.295
U120B 100000 120000 0.146 0.255 3.9 0.295
U160BS 110000 160000 0.146 0.28 6.2 0.315
U160BL 110000 160000 0.17 0.28 6.2 0.34
U210B 110000 210000 0.17 0.3 7.2 0.37
U300B 130000 300000 0.195 0.33 10 0.39

6.2 Selection criteria

The selection of the appropriate insulators for suspension and angle towers requires two main
criterion, one electrical and another mechanical. Both criteria should be respected for the MV
and HV voltage levels.

6.2.1 Electrical criterion

For the electrical criterion, the selected insulator should withstand the electrical
requirements for a certain valid length of the string. For both types of insulators, the
necessary number of elements to form an insulator string is calculated such that the total
elements withstand the maximum voltage under normal conditions, under wet conditions, and
under lightning impulse. The calculation of the number of elements for each condition is
presented using Equations (6.1), (6.2) and (6.3).

Number of elements under normal conditions:

𝑛𝑛𝑜𝑟𝑚𝑎𝑙 ⪖ 𝑈𝑠 · 𝜖0 (6.1)
𝜖

Where:

• 𝑛𝑛𝑜𝑟𝑚𝑎𝑙 is the number of elements necessary to withstand maximum voltage under normal
conditions
• 𝑈𝑠 is the maximum voltage of the system [𝑉 ]
−6
• 𝜖0 is the minimum nominal creepage for a medium level of polution being 20 · 10 accord-
ing to IEC [11] and [16] [𝑚/𝑉 ]
• 𝜖 is the insulator creepage distance [𝑚]

Number of elements under wet conditions:

𝑈𝑝 𝑓
𝑛 𝑤𝑒𝑡 ⪖ (6.2)
𝑈𝑤

Overhead line 2
 Chapter 6.

Where:

• 𝑛 𝑤𝑒𝑡 is the number of elements necessary to withstand maximum voltage under wet con-
ditions
• 𝑈𝑝 𝑓 is the maximum power frequency voltage [𝑉 ]
• 𝑈𝑤 is the long term wet voltage of the insulator [𝑉 ]

Number of elements under lightning impulse:

𝑈𝑓 𝑓
𝑛𝑖𝑚𝑝𝑢𝑙𝑠𝑒 ⪖ (6.3)
𝑈𝑙

Where:

• 𝑛𝑖𝑚𝑝𝑢𝑙𝑠𝑒 is the number of elements necessary to withstand maximum voltage under light-
ning impulse
• 𝑈𝑓 𝑓 is the maximum lightning withstand voltage [𝑉 ]
• 𝑈𝑙 is the maximum lightning voltage of the insulator [𝑉 ]

The number of elements composing the insulator string is the biggest number of elements
among the three conditions calculated above. On the other hand, the insulator string must
comply with the minimum clearance phase to earth to guarantee the clearances respect for
all parts of the tower. To validate the insulator length a safety factor is applied to the
minimum phase-earth clearance to consider possible swing angle, then the length of the
insulator is calculated using Equation (6.4).

𝐿𝑖𝑛𝑠 = 𝑛𝑒𝑙𝑒 · 𝑙𝑒𝑙𝑒 (6.4)

Where:

• 𝐿𝑖𝑛𝑠 is the length of the insulator [𝑚]

• 𝑛𝑒𝑙𝑒 is the total number of elements composing the insulator string
• 𝑙𝑒𝑙𝑒 is the length of the insulator’s element [𝑚]

The insulator length calculated above, should be higher than the minimum length defined by
the phase to earth clearance such as: 𝑑𝑒𝑙 1.1 for the insulator to be electrically valid. Once the
·
electrical insulator is selected, the insulator parameters such as the length, weight or minimum
creepage are calculated upon the number of its elements.

6.2.2 Mechanical criterion

Mechanically, the insulator must withstand the maximum load of the conductors to have the
optimal choice of insulator sets; based on this criterion, different combinations of insulators per
conductor bundle are tested. For every combination, the insulator strength against the conduc-
tor’s bundle is checked; if the insulator withstands the bundle’s load, the number of insulators
set is kept. If it doesn’t withstand the load, a new configuration with more insulators is tried.
The possible sets are simplex with one insulator, duplex with two, or quadruplex with four
insulators per conductor(s).

Overhead line 3
 Chapter 6.

From the possible insulator combinations, the insulator with the closest minimum failing load
to the expected tensile strength is selected, this latter is the typical tensile strength of insulators
depending on the voltage. In Table 6.3 the expected tensile strength is chosen based on several
study cases from Spain.

Table 6.3: The expected tensile strength for insulators

Maximum voltage 𝑈 [kV] Expected tensile strength [N]

⩽36 70000
⩽245
⩽72.5 160000
120000
> 245 210000

Ultimately, the insulators that have a minimum failing load closest one to the expected tensile
strength are the candidates that respect both the electrical and mechanical criteria. However,
there is a possibility to get two insulators with same tensile strength, and to select the appropri-
ate one, the insulator with the least length and creepage distance is prioritized by giving more
importance to minimizing the length.

Overhead line 3
 Chapter 7.

Chapter 7

Clearances

There are several types of electrical clearances, some are determined to prevent disruptive dis-
charges between the conductor and the earthed tower, others to prevent disruptive discharges
between phase conductors. In addition, there are mid-span clearances that need to be respected
during wind conditions and other safety clearances to obstacles or possible objects in the path.

In this chapter the electrical clearances covered in this methodology will be described.

7.1 Electrical clearances

The electrical clearances are necessary to design the top-geometry of the towers to ensure the
line’s reliability. These internal phase to phase and phase to earth clearances are calculated
based on the required withstand voltage and the statistical switching over-voltage, both
calculated from the insulation coordination procedure. Chapter 5

For each over-voltage stress, the required withstand voltage is calculated and used to determine
the required electrical clearances phase-phase and phase-earth. Following, the clearances for-
mulas are presented for each voltage stress.

After the calculation of phase to phase and phase to earth clearances for each over-voltage, the
maximum clearances among them are chosen and compared against standard clearances accord-
ing to EN 50341 [17]. Based on the voltage of the system, as listed in the Table 7.1, standard
phase to phase and phase to earth clearances are selected. The maximum between the clearances
calcu- lated and the standard values are the final minimum clearances to be respected when
designing the towers.

Overhead line 3
 Chapter 7.

Table 7.1: The minimum standard clearances pp and pe. Source: EN standard [17]

Maximum voltage Phase-earth Phase-

phase
3.6 0.08 0.1
12
7.2 0.12
0.09 0.15
0.1
17.5 0.16 0.2
24 0.22 0.25
30 0.27 0.33
36 0.35 0.4
72.5
52 0.7
0.6 0.8
0.7
82.5 0.75 0.85
100 0.9 1.05
123 1.0 1.15
145 1.2 1.4
245
170 1.7
1.3 2.0
1.5
300 2.1 2.4
420 2.8 3.2
525 3.5 4.0
765 4.9 5.6

7.1.1 Phase to phase clearances

The phase-phase clearance is the minimum distance that must be respected between the phase
conductors and it is calculated for each stress as follows:

Fast-front overvoltages

1.2 · 𝑈 90% 𝑓 𝑓 −𝑖𝑛𝑠

𝐷𝑝𝑝 − 𝑓 𝑓 = 530 · 𝐾 · 𝐾 (7.1)
𝑎 𝑧 − 𝑓 𝑓 · 𝐾𝑔 − 𝑓 𝑓

Where:

• 𝐷 𝑝𝑝 − 𝑓 𝑓 is the phase to phase clearance with fast-front overvoltages [𝑚]

• 𝑈 90%𝑓 𝑓 −𝑖𝑛𝑠 is the lightning withstand voltage of the insulator [𝑘𝑉 ]
• 𝐾𝑎 is the altitude factor calculated using table 2.15 found in [3]
• 𝐾𝑧−𝑓 𝑓 is the deviation factor for fast-front overvoltages being 0.961 according to [3]
• 𝐾𝑔−𝑓 𝑓 is the gap factor for fast-front overvoltages being 1.16 according to [3]

Slow-front overvoltages

𝐷𝑝𝑝 −𝑠 𝑓
1.4 · 𝐾𝑐𝑠 · 𝑈2%𝑠 𝑓
1080 · 𝐾𝑎 · 𝐾𝑧−𝑠 𝑓 · 𝐾𝑔−𝑠𝑓 = 2.17 · −1
(7.2)

Wher

Overhead line 3
 Chapter 7.

• 𝐷𝑝𝑝−𝑠 𝑓 is the phase to phase clearance with slow-front overvoltages [𝑚]

• 𝐾𝑐𝑠 is the statistical coordination factor being 1.05 [3]
• 𝑈2%𝑠 𝑓 is the statistical switching overvoltage [𝑘𝑉 ]
• 𝐾𝑎 is the altitude factor using table 2.15 found in [3]
• 𝐾𝑧−𝑠𝑓 is the deviation factor for slow-front overvoltages being 0.922 according to [3]
• 𝐾𝑔−𝑠 𝑓 is the gap factor for slow-front overvoltages being 1.6 according to [3]

Power frequency voltages

𝑈 0.833
−
𝑒𝑥𝑝 1
𝐷𝑝𝑝−𝑝𝑓 = 1.64 · 750 · 𝐾 · �𝐾𝑧 −𝑝 𝑓 · 𝐾𝑔−𝑝 𝑓 (7.3)
Where:

• 𝐷𝑝𝑝−𝑝𝑓 is the phase to phase clearance with power frequency voltages [𝑚]
• 𝑈 is the maximum voltage of the system [𝑘𝑉 ]
• 𝐾𝑎 is the altitude factor
• 𝐾𝑧−𝑝 𝑓 is the deviation factor for power frequency voltages being 0.910 according to [3]
• 𝐾𝑔−𝑝𝑓 is the gap factor for power frequency voltages being 1.26 according to [3]

7.1.2 Phase to earth clearances

The phase to earth clearance is the minimum distance to be respected between a phase
conductor and both an earthed part of the tower and the earth wire, and it is calculated for each
stress voltage as follows:

Fast-front overvoltages

𝑈 90% 𝑓 𝑓 −𝑖𝑛𝑠
𝐷𝑝𝑒 − 𝑓 𝑓 = (7.4)
530 · 𝐾𝑎 · 𝐾𝑧 − 𝑓 𝑓 · 𝐾𝑔− 𝑓 𝑓

Where:

• 𝐷𝑝𝑒−𝑓 𝑓 is the phase to earth clearance with fast-front overvoltages [𝑚]

• 𝑈 90%𝑓 𝑓 −𝑖𝑛𝑠 is the lightning withstand voltage of the insulator [𝑘𝑉 ]
• 𝐾𝑎 is the altitude factor
• 𝐾𝑧−𝑓 𝑓 is the deviation factor for fast-front overvoltages being 0.961 according to [3]
• 𝐾𝑔−𝑓 𝑓 is the gap factor for fast-front overvoltages being 1.12 according to [3]

Slow-front overvoltages

𝐷𝑝𝑒 −𝑠 𝑓 𝐾𝑐𝑠 · 𝑈2%𝑠𝑓

1080 · 𝐾𝑎 · 𝐾𝑧−𝑠𝑓 · 𝐾𝑔−𝑠 𝑓 = 2.17 · 𝑒𝑥𝑝 −1
(7.5)
Wher

Overhead line 3
 Chapter 7.

• 𝐷𝑝𝑒−𝑠 𝑓 is the phase to earth clearance with slow-front overvoltages [𝑚]

• 𝐾𝑐𝑠 is the statistical coordination factor being 1.05 [3]
• 𝑈2%𝑠 𝑓 is the statistical switching overvoltage [𝑘𝑉 ]
• 𝐾𝑎 is the altitude factor
• 𝐾𝑧−𝑠𝑓 is the deviation factor for slow-front overvoltages being 0.922 according to [3]
• 𝐾𝑔−𝑠 𝑓 is the gap factor for slow-front overvoltages being 1.45 according to [3]

Power frequency voltages

"
𝑈
# 0.833
√ −
𝑒𝑥𝑝
750 · 3 · 𝐾 · 𝐾𝑧 −𝑝 𝑓 · 𝐾𝑔−𝑝 𝑓
�
𝐷𝑝𝑒−𝑝 𝑓 = 1.64 · 1
(7.6)
Where:

• 𝐷𝑝𝑝−𝑝𝑓 is the phase to earth clearance with power frequency voltages [𝑚]
• 𝑈 is the maximum voltage of the system [𝑘𝑉 ]
• 𝐾𝑎 is the altitude factor
• 𝐾𝑧−𝑝 𝑓 is the deviation factor for power frequency voltages being 0.910 according to [3]
• 𝐾𝑔−𝑝𝑓 is the gap factor for power frequency voltages being 1.26 according to [3]

7.2 Mid-span clearances

Designing a reliable tower is not limited only to the electrical clearances, but also to the mid-
span clearances, which are the distances between the phase conductors in the middle of a span
to avoid flash-overs under extreme wind actions.

Therefore, to calculate the mid-span clearances, the swing angle of the conductor is calculated
to determine the most unfavorable position of the phase conductors in the span.

Swing angle
The swing angle is calculated to consider possible wind actions on the conductors and
insulators; in this methodology, it is calculated considering an extreme wind speed. The swing
angle defines a swung position of the conductors that determines the mid-span clearance.
Consequently, using Equations (7.7) and (7.10) the swing angles of either a phase conductor or
an insulator with a conductor are calculated for a wind speed of 120 km/h. Figure 7.1 shows an
illustration of a swing angle of an insulator.

Overhead line 3
 Chapter 7.

Figure 7.1: The swing angle

Swing angle for a conductor:

"
𝜙𝑐 = 𝑡𝑎𝑛𝑔 −1 (𝜌/2) · 𝐶𝑐 · 𝑉 2 · 𝐺𝐿 · 𝐷 · 𝑎𝑤
(7.7)
𝑚𝐶 · 𝑔 �· 𝑎𝑤𝑔

Where:

• 𝜙𝑐 is the swing angle for a conductor in degrees

• 𝜌 is the air density depending on temperature, humidity and altitude calculated using
Equa- tion (7.8) [𝑘𝑔/𝑚3]
• 𝐶𝑐 is the drag factor equal to 1.0
• 𝑉𝑅 is the reference wind speed taken as 33.3 [𝑚/𝑠]
• 𝐺𝐿 is the span factor taking into account the effect of wind span, it is calculated in
Equation (7.9)
• 𝐷 is the diameter of the conductor [𝑚]
• 𝑎𝑤 is the wind span [𝑚]
• 𝑎𝑤𝑔 is the weight span [𝑚]
• 𝑚𝑐 is the conductor mass per unit length [𝑘𝑔/𝑚]
• 𝑔 is the gravity 9.81 [𝑚/𝑠 2 ]

The air density is calculated using the following equation according to [3] :

298
𝜌 = 𝜌0 𝑇+ 𝑒𝑥𝑝 (−0.00012 · 𝐻 ) (7.8)

Where:

• 𝜌 is the air density [𝑘𝑔/𝑚3]

Overhead line 3
 Chapter 7.

• 𝑇 is the ambient temperature in Celsius

·
• 𝜌 0 is the density at 15 C equal to 1.225 [𝑘𝑔/𝑚 3 ]
• 𝐻 is the altitude above sea level [𝑚]

The span factor used to calculate the swing angle is calculated according to the IEC standard
[18] as follows:

−10 −7 −4
𝐺𝐿 = (4 · 10 · 𝑎3 ) − (5 · 10 · 𝑎2 ) − (10 · 𝑎𝑤) + 1.0403 (7.9)
𝑤 𝑤

Where:

• 𝐺𝐿 is the span factor

• 𝑎𝑤 is the wind span [𝑚]

Swing angle for an insulator:

"
(𝜌/2) · 𝐶𝑐 · 𝑉 2 · 𝐺𝐿 · 𝐷 · 𝑎𝑤 + 𝑄𝑊 𝑖𝑛𝑑 /2
𝜙𝑖𝑛𝑠 = 𝑡𝑎𝑛𝑔 −1 � (7.10)
𝑊𝐶 + 𝑊𝑖𝑛𝑠 /2

Where:

• 𝜙𝑖𝑛𝑠 is the swing angle for an insulator in degrees

• 𝜌 is the air density depending on temperature, humidity and altitude calculated using
Equa- tion (7.8) [𝑘𝑔/𝑚3]
• 𝐶𝑐 is the drag factor equal to 1.0
• 𝑉𝑅 is the reference wind speed taken as 33.3 [𝑚/𝑠]
• 𝐺𝐿 is the span factor taking into account the effect of wind span, calculated in Equation
(7.9)
• 𝐷 is the diameter of the conductor [𝑚]
• 𝑎𝑤 is the wind span [𝑚]
• 𝑄𝑊 𝑖𝑛𝑠 is the wind load on the insulator set calculated using Equation (7.11) [𝑁 ]
• 𝑊𝑐 is the effective conductor weight [𝑁 ]
• 𝑊𝑖𝑛𝑠 is the dead weight of the insulator set [𝑁 ]

The wind load on the insulator set is calculated according to the IEC standard [18]

𝑄𝑊 𝑖𝑛𝑠 = 𝑃 · 𝐶𝑖𝑛𝑠 · 𝐴𝑖𝑛𝑠 (7.11)

Where:

• 𝑄𝑊 𝑖𝑛𝑠 is the wind load on the insulator set [𝑁 ]

• 𝑃 is the wind pressure calculated using Equation (7.12) [𝑃𝑎]
• 𝐶𝑖𝑛𝑠 is the drag factor of the insulator being 1.2

Overhead line 3
 Chapter 7.

• 𝐴𝑖𝑛𝑠 is the area exposed by the insulator set; the area is considered as a rectangle multi-
plying the insulator diameter with its length [𝑚𝑚]

The wind pressure is calculated based on the air density according to the IEC 60826 standard
[18]:

𝑃 = 1/2 · 𝜌 · 𝑉 2 (7.12)
�

Where:

• 𝑃 is the wind pressure [𝑃𝑎]

• 𝜌 is the air density depending on temperature, humidity and altitude calculated using
Equa- tion (7.8) [𝑘𝑔/𝑚3]
• 𝑉𝑅 is the reference wind speed taken as 33.3 [𝑚/𝑠]

Calculating the swing angle for a conductor, allows the calculation of the mid-span clearances,
both phase-phase and phase-earth. [3]

Phase-phase midspan clearance:

The phase to phase mid-span clearance is the distance between two parallel phase conductors
and it is calculated as follows:

√︁
𝑐𝑚𝑖𝑛 −𝑝𝑝 = 𝐾𝑐 𝑓𝑐 + 𝑙𝑘 + 0.75 · 𝐷𝑝𝑝 (7.13)

Where:

• 𝑐𝑚𝑖𝑛 −𝑝𝑝 is the phase to phase mid-span clearance [𝑚]

• 𝐾𝑐 is the swinging coefficient determined based on the swing angle in Table 7.2
• 𝑓𝑐 is the sag of the conductor [𝑚]
• 𝑙𝑘 is the length of the insulator set swinging orthogonally to the line direction [𝑚]
• 𝐷𝑝𝑝 is the phase-phase electrical clearance calculated using Equation (7.1) [𝑚]

Table 7.2: The minimum standard clearances pp and pe [Source EN]

Swing angle Swinging coefficient
𝜙 [◦] 𝐾𝑐
⩽40 0.6
⩽55 0.62
⩽65 0.65
>65 0.7
Phase-earth midspan clearance:

The phase to earth mid-span clearance is the distance between the phase conductors and the
earth wire and it is calculated as follows:

Overhead line 3
 Chapter 7.

√︁
𝑐𝑚𝑖𝑛 −𝑝𝑒 = 𝐾𝑐 𝑓𝑐 + 𝑙𝑘 + 0.75 · 𝐷𝑝𝑒 (7.14)

Where:

• 𝑐𝑚𝑖𝑛 −𝑝𝑒 is the phase to earth mid-span clearance [𝑚]

• 𝐾𝑐 is the swinging coefficient determined based on the swing angle in Table 7.2
• 𝑓𝑐 is the sag of the conductor [𝑚]
• 𝑙𝑘 is the length of the insulator set swinging orthogonally to the line direction [𝑚]
• 𝐷𝑝𝑒 is the phase-earth electrical clearance calculated using Equation (7.4) [𝑚]

Finally, after calculating the phase to phase and phase to earth clearances at the middle of the
span, the mid-span clearance of each tower is considered as the maximum between those of its
previous and next spans.

7.3 Safety distances

As mentioned at the beginning of this chapter, some safety distances are determined to ensure a
reliable design of the overhead line. Among these safety distances, the minimum ground clear-
ance and the distance to object as seen in Figure 7.2 are determined.

Figure 7.2: Safety distances of the line

Minimum ground clearance:

The minimum ground clearance is the minimum distance to the ground that all the catenaries in
the line’s spans must respect. This distance is calculated considering the electrical phase to earth
clearance and it is considered from the lowest point of the conductor catenary. Accordingly, the
minimum ground clearance is calculated using Equation (7.15).

𝑑𝑚𝑖𝑛 = 𝐷𝑝𝑒 + 𝑎𝑎𝑑𝑑 (7.15)

Overhead line 3
 Chapter 7.

Where:

• 𝑑𝑚𝑖𝑛 is the minimum distance to the ground [𝑚]

• 𝐷𝑝𝑒 is the phase-earth electrical clearance calculated using Equation (7.4) [𝑚]
• 𝑎𝑎𝑑𝑑 is the additional safety distance according to [3] being 5 [𝑚]

Minimum distance to object:

The minimum distance to object is the minimum distance that a user should consider in case of
crossing an object such as a building in the line’s path. This distance is calculated using
Equation (7.16) as follows:

𝐷𝑎𝑑𝑑 = 𝐷𝑝𝑒 + 𝑎𝑎𝑑𝑑 (7.16)

Where:

• 𝐷𝑎𝑑𝑑 is the minimum distance to the ground [𝑚]

• 𝐷𝑝𝑒 is the phase-earth electrical clearance calculated using Equation (7.4) [𝑚]
• 𝑎𝑎𝑑𝑑 is the additional safety distance according to [3] being 10 corresponding to a non-fire
resistant roof [𝑚]

Overhead line 4
 Chapter 8. Tower

Chapter 8

Tower geometry

The top-geometry of the line’s towers is determined based on all the parameters explained in the
previous chapters. The clearances between the conductors, the clearances between the conductor
and the earthed parts of the tower, the mid-span clearances, and the effect of the wind action on
the conductor and the insulator are all considered when designing a tower’s top-geometry to
ensure the reliability of the line.

In this chapter the line’s towers top-geometry calculation is explained and the different param-
eters that define the tower’s parts are described.

8.1 General procedure

After selecting an adequate tower based on the voltage and capacity as explained in Chapter 4,
the top-geometry of the tower is determined based on the electrical clearances. In order to define
the final geometries for each type of tower designed in this methodology, the following
procedure is followed:

• The tower is defined with nodes that represent the possible connections of the conductor
and insulator with the tower’s cross-arms and body, an example of these nodes is
illustrated in Figure 8.1.
• The coordinates of each node are determined based on the electrical and mid-span clear-
ances calculated previously
• Based on the tower category, either suspension or angle, some parameters such as the
swing angle is calculated to define the peak distance
• From the nodes’ coordinates, the tower’s components and distances are calculated:

– the tower’s cross-arm distance

– the peak distance
– the distance between the cross-arms vertically
– the width of all the towers’ body is considered as 1m.
• Based on the tower type, the top-geometry calculated is ,then, compared with a standard
catalog of towers [19].

Overhead line 4
 Chapter 8. Tower

• If a standard tower that meets this methodology’s design criteria is found, it gets selected,
if not, the tower calculated based on the procedure above is considered instead.

Figure 8.1: Top-geometry of an S tower

8.2 Peak distance calculation

The peak distance as shown in Figure 8.1 is the vertical distance between the highest phase
conductor to the earth wire, considering the swing angle in the most unfavorable conditions.
The peak distance is calculated using Equation (8.1) depending on the tower type that defines
the position of the insulator; if it is a suspension or tension insulator.

Moreover, this distance is compared against the phase-earth midspan clearance calculated pre-
viously to consider the highest distance and ensure the respect of all the clearances.

𝐷𝑝𝑒 + (𝑙𝑖𝑛𝑠 · 𝑠𝑖𝑛 (𝜙 ) )

𝐷𝑝𝑒𝑎𝑘 = (8.1)
𝑡𝑎𝑛(𝛼)

Where:

• 𝐷𝑝𝑒𝑎𝑘 is the peak distance [𝑚]

• 𝐷𝑝𝑒 is the electrical phase-earth distance calculated using Equation (7.4) [𝑚]
• 𝑙𝑖𝑛𝑠 is the insulator set length [𝑚]
• 𝜙 is the swing angle in degrees
◦
• 𝛼 is the shield angle of the earth wire in degrees, being 30

Overhead line 4
 Chapter 8. Tower

8.3 Standard towers top-geometry

As explained in the general procedure of calculating the tower top-geometry, the selection of
a standard tower is possible when it respects the design criteria. In this methodology, similar
towers from Imedexsa’s catalog [19] to the designed towers were selected, based on the shape,
the voltage and the number of circuits.

The dimensions of the standard towers listed in tables 8.1 to Table 8.5 are used to select the
final tower’s top-geometry for the MV and HV voltage levels.

Table 8.1: Tower type Atorinillada N shape. [Source: Imedexsa [19]]

Type a b c h
N0 1 1.2 1.25 1.5
N1 1.25 1.2 1.5 1.5
N2 1.5 1.8 1.75 1.5

Table 8.2: Tower type Condra S shape. [Source: Imedexsa [19]]

Type a b c h
S3 3 3.3 3.2 –
S4 4.1 4.4 4.3 –
S5 4.1 5.5 3.3 –
S3C 3 3.3 3.2 4.3
S4C 4.1 4.4 4.3 5.9
S5C 4.1 5.5 3.3 5.9

Table 8.3: Tower representing the Pi tower. [Source: Imedexsa [19]]

Type a b c h
D3 3 13.5 7.5 4.3
D5 3.2 15 8.6 4.3
D7 3.8 17 9.4 5.5

Table 8.4: Tower type Condor N Doble. [Source: Imedexsa [19]]

Type b a/c d-e

1 3.3 3 4.3
2 4.4 3.2 5.2
3 5.5 3.6 5.9
54 –– 4.1
3.8 3.3-3
6.6
6 – 4.3 4.4-
7 – 4.6 5.5-3
8 – 4.9 –

Overhead line 4
 Chapter 8. Tower

Table 8.5: Tower type Icaro N shape. [Source: Imedexsa [19]]

Type b a/c d-e

1 5.8 4.5 7.2
32 –– 65 4.9-3.5
8.6
4 – 6.5 6.2-

Overhead line 4
 Chapter 9. Electrical

Chapter 9

Electrical calculation

The main purpose of an overhead line is to carry the electrical energy from the generation point
to the delivery point. Therefore, calculating the electrical characteristics of the line is essential
in order to ensure a reliable transmission of the energy. A poorly designed overhead line from
an electrical perspective could cause instability and excessive losses.

The quality of the design will depend on the conductor selection and the circuit arrangement,
which will determine the electrical parameters of the line. This chapter presents the basic formu-
lae for calculating the electrical parameters of the line, as well as the voltage drop, power losses
and power factor.

9.1 Electrical models

When calculating the electrical behavior of an overhead line, the literature proposes three dif-
ferent calculation methods with different complexity and accuracy. The selection of the model
typically depends on the length of the overhead line.

• Short-Line model: The equivalent circuit shown in Figure 9.1 only takes into account the
effect of the resistance and the reactance of the conductor, disregarding the effect of the
capacitance of the line. The Short-Line model is only used for lengths up to 80 km, losing
accuracy for longer lines.

Figure 9.1: Equivalent circuit of the short line model

• Pi Model: Takes into account the effect of the capacitance on the overhead line. This
model in Figure 9.2 considers that the electrical parameters of the line are concentrated
and not distributed along the line, with better accuracy than the one provided by the
Short-Line

Overhead line 4
 Chapter 9. Electrical

model, but the complexity of the calculation increases. The Pi model is used for lines with
a length up to 300 km.

Figure 9.2: Equivalent circuit of the pi model

• Distributed Parameters Model This model represents the exact mathematical model for
an overhead line with evenly distributed parameters across the line, providing accurate
results even for lengths over 300 km. The distributed parameters model can be
represented by either exponential functions or hyperbolic functions and the equivalent
circuit is shown in Figure 9.3

Figure 9.3: Equivalent circuit of the distributed parameters model

9.1.1 Distributed Parameters Model

After analyzing the advantages and disadvantages of the three electrical models, the distributed
parameters model was selected due to its accuracy, especially for longer lines. Therefore, this
section will be referring to the equivalent circuit shown in Figure 9.3.

The model is based on four electrical parameters: resistance (R), inductance (X), susceptance
(B) and conductance (G), which will be discussed in later sections of the chapter. A typical
simplifi- cation that is done is to not consider the conductance G due to its residual effect on
the result.

In order to calculate the behavior of the circuit, the differential equations of the electrical model
have to be solved, knowing the values of the voltage and current at the sending end (substation
point) Vs and Is.

Equations (9.1) and (9.2) represent the differential equations of the distributed parameters model:
𝑑𝑈 𝑑𝑥
−→ = (𝑅 + 𝑗𝑋 ) · 𝐼→

Overhead line 4
 Chapter 9. Electrical

(9.1)

Overhead line 4
 Chapter 9. Electrical

→
𝑑𝐼
– = (𝐺 + 𝑗𝐵) · 𝑈→ (9.2)
�
From the previous expressions, the following equations can be deduced [3] :
! !
→ − → · → → + → · →
𝑈 𝑠 𝐼 𝑠 𝑍 0 𝑈 𝑠 𝐼 𝑠 𝑍 0
𝑈→ = · exp (𝛾 · 𝑥 ) + · exp (−𝛾 · 𝑥 ) (9.3)
2 2
! !
→ → → → → →
𝐼 𝑠 − 𝑈 𝑠 /𝑍 0 𝐼 𝑠 + 𝑈 𝑠 /𝑍 0
𝐼→ = 2 · exp (𝛾 · 𝑥 ) + 2 · exp (−𝛾 · 𝑥 ) (9.4)

Where:

• 𝑈→ is the phase to earth voltage at the receiving point

• 𝐼→ is the current at the receiving point
• 𝑈→𝑠 is the phase to earth voltage at the sending point
• 𝐼→𝑠 is the current at the sending point
• 𝛾 is the propagation constant of the line which is defined as
√︁
𝛾 = (𝑅 + 𝑗𝑋 ) · (𝐺 + 𝑗𝐵) (9.5)

• 𝑍→0 is the characteristic impedance of the line which is defined as

√
𝑅 + 𝑗𝑋
→
𝑍 0 = ︄ 𝐺 + 𝑗𝐵 (9.6)

Equations (9.3) and (9.4) represent the distributed parameters model using exponential
equations. However, there is another way to represent the equations using hyperbolic functions,
that derives in the following expressions [3]:

𝑈→ = 𝑈→𝑠 · cosh 𝛾 − 𝐼→𝑠 · 𝑍→0 · sinh 𝛾 (9.7)

𝐼→ = 𝐼→𝑠 · cosh 𝛾 𝑈→𝑠

sinh 𝛾 (9.8)
–
· 0
𝑍→
9.2 Electrical Parameters
The aforementioned electrical model presents an electrical circuit with different components,
which are used to calculate the voltage and current at the receiving end of the overhead line.
Even though the distributed parameters model is very accurate, the electrical parameters used
in it must be estimated accordingly in order to obtain a reliable result.

It is important to note that the electrical parameters will not only depend on the conductor’s
material and composition, but also on their geometrical disposition throughout the line.

Overhead line 4
 Chapter 9. Electrical

9.2.1 Geometrical Mean Distance

The geometrical mean distance (GMD) is the equivalent distance between the conductor
bundles of an overhead line, and will affect directly the values of the inductance and the
capacitance of the overhead line.

The calculation of the GMD depends on the number of circuits installed in the line. This section
only covers the calculation of the GMD of configurations with one circuit and with two circuits,
which are the most widely used configurations.

In Figure 9.5, an example configuration of a one circuit configuration is shown, with the three
phase conductors separated by distances 𝐷𝑎𝑏, 𝐷𝑎𝑐 𝑎𝑛𝑑 𝐷𝑏𝑐 .

Figure 9.4: Phases of a tower with one circuit for the GMD calculation

The GMD in this case can be calculated according to Equation (9.9) , where Dij refers to the
distance between phase i and phase j :

√︁
𝐺𝑀𝐷 = 𝐷𝑎𝑏 · 𝐷𝑎𝑐 · 𝐷𝑏𝑐
3
(9.9)

The two circuits configuration is shown in Figure 9.5, where the first circuit is formed by con-
ductors a, b and c and the second circuit is formed by conductors a’, b’ and c’.

Overhead line 4
 Chapter 9. Electrical

Figure 9.5: Phases of a tower with two circuits for the GMD calculation

In this case, the GMD will be calculated according to Equations (9.10), (9.11), (9.12) and (9.13):

√︁
𝐷𝐴𝐵 = 𝐷𝑎𝑏 · 𝐷𝑎𝑏 ′ · 𝐷𝑎′𝑏 · 𝐷𝑎′𝑏 ′
4
(9.10)
√︁
4
′ ′ ′ ′𝐷 =
√︁
𝐷𝐶𝐴𝐷= 𝐷𝑐𝑎· ·𝐷𝐷𝑐𝑎′ ·· 𝐷
𝐷𝑐′𝑎 · 𝐷𝑐′𝑎′ ·𝐷
4
(9.12)
𝐺𝑀𝐷 = 𝐷𝐴𝐵 · 𝐷𝐵𝐶 · 𝐷𝐶𝐴
3
(9.13)
√︁
Another GMD that will be used in the inductance calculation is the one that establishes a rela-
tionship between the same phases of both circuits, defined by

√︁
𝐺𝑀𝐷𝑝𝑝 = 𝐷𝑎𝑎′ · 𝐷𝑏𝑏 ′ · 𝐷𝑐𝑐 ′
3
(9.14)

The previous expressions calculate the GMD of the circuits at a specific point of the overhead
line. However, the conductor’s disposition will change throughout the line, resulting in different
values for GMD for different points in the line. In order to obtain a more accurate model, an
average of the GMDs across the overhead line is calculated.

The average GMD within a span can be obtained from the GMD of its neighbor towers, accord-
ing to Equation (9.15) , with GMD1 and GMD2 being the GMD at the previous and next towers
respectively. This operation can be performed because the variation of the GMD across the span
is linear.

𝐺 𝑀 𝐷1 +
𝐺𝑀𝐷 𝑠𝑝𝑎𝑛 = (9.15)
2
𝐺 𝑀 𝐷2

Therefore, the average GMD of the entire overhead line can be calculated as a weighted average,

Overhead line 5
 Chapter 9. Electrical

considering the span length as the weight of the function:

Overhead line 5
 Chapter 9. Electrical

.𝑠𝑝𝑎𝑛𝑠
𝑖=1
𝐺𝑀𝐷 𝑜ℎ𝑙 = (9.16)
𝐿𝑜ℎ𝑙

Where: 𝐺𝑀 · 𝐿𝑖 𝑖

• 𝐺𝑀𝐷 𝑜ℎ𝑙 is the average GMD of the entire overhead line [m]
• 𝐺𝑀𝐷 𝑖 is the average GMD of span i [m]
• 𝐿𝑖 is the length of span i [m]
• 𝐿𝑜ℎ𝑙 is the sum of the length of all the spans in the overhead line [m]

Similarly, 𝐺𝑀𝐷𝑝𝑝 can also be calculated as a weighted average throughout the overhead line
with Equation (9.17) , in which 𝐺𝑀𝐷 𝑝𝑝𝑖 is the average GMDpp in a span, calculated from
Equation (9.15)
.𝑠𝑝𝑎𝑛𝑠
𝑖=1 𝑖 · 𝐿𝑖
𝐺𝑀𝐷 𝑝𝑝𝑜ℎ𝑙 = (9.17)
𝐺𝑀𝐷𝑝𝑝𝐿𝑜ℎ𝑙

9.2.2 Geometrical Mean

Radius
The Geometrical Mean Radius (GMR) is the average of the distances between the conductors
within the same bundle, which will affect the calculation of the inductance and capacitance
across the overhead line.

Figure 9.6 shows a bundle of four conductors with a separation d between them (the separation
will be 400 mm by default for the calculation model).

Figure 9.6: Four conductors bundle

Equation (9.18) shows the general expression to calculate the GMR in a symmetrical configura-
tion: 𝐺𝑀𝑅 =
𝑛
v, .
𝑑 1→𝑖 (9.18)
𝑖=2
𝑛
𝑟′·
Where:

• 𝑟 ′ is the GMR of a single conductor, defined as 𝑟 ′ = exp 0.25 𝑟 with r being the radius
of the conductor [m] (−
• 𝑛 is the number of conductors in the bundle

Overhead line 5
 Chapter 9. Electrical

• 𝑑 1→𝑖 is the distance between the first conductor and conductor i [m]

For the capacitance calculation, the radius r’ will be substituted by the actual radius of the con-
ductor r, as shown in Equation (9.19)
v, 𝑛
.
𝑑 1→𝑖 (9.19)
𝐺𝑀𝑅𝐶 = 𝑛
𝑟 · 𝑖=1

The previous equation can be rewritten for different number of conductors in the bundle, where
d is the separation between conductors and r is the radius of the conductor.

𝐺𝑀𝑅1 = 0.7788 · 𝑟 (9.20) 𝐺𝑀𝑅1𝐶 = 𝑟 (9.21)

√ √
𝐺𝑀𝑅2 = 0.8825 · 𝑟 · 𝑑 (9.22) 𝐺𝑀𝑅2𝐶 = 𝑟 · 𝑑 (9.23)
√ √
3 3
𝐺𝑀𝑅3 = 0.9200 · 𝑟 · 𝑑2 (9.24) 𝐺𝑀𝑅3 � = 𝑟 · 𝑑2 (9.25)
√ √
4 4
𝐺𝑀𝑅4 = 1.0244 · 𝑟· 𝑑3 (9.26) 𝐺𝑀𝑅4 � = 1.0905 · 𝑟 · 𝑑3 (9.27)
The GMR throughout the overhead line will remain constant, as the bundle disposition will not
change at any point of the line. Therefore, there is no need to calculate an average in this case.

9.2.3 Resistance
The resistance of the overhead line is one of the most influencing parameters due to its direct
dependency with the power losses. A high resistance value will yield high power losses,
resulting in a poor design of the line. On the other hand, low resistance conductors have larger
sections, increasing the costs and the mechanical loads.

In order to compute the resistance of the entire overhead line, the calculated AC resistance from
Equation (2.1) will be used, as it represents the AC resistance of a single conductor.

The equation that is used to calculate the resistance of the entire overhead line is the following:
𝑅𝐴𝐶
𝑅= 𝑙 (9.28)
𝑛𝑐 · 𝑛𝑠 ·
Where:

• 𝑅 is the resistance of the entire line [Ω]

• 𝑅𝐴𝐶 is the AC resistance of a single conductor [Ω/m]
• 𝑛𝑐 is the number of circuits
• 𝑛𝑠 is the number of subconductors
• 𝑙 is the length of the line [𝑚]

As it can be seen from Equation (9.28), the resistance of the overhead line has a linear
dependency on the number of conductors of the line.

9.2.4 Inductance and Reactance

A current carrying conductor will produce a magnetic field around itself, resulting in an in-

Overhead line 5
 Chapter 9. Electrical

ductance in AC systems due to the variation of the conductor’s current. When more than one

Overhead line 5
 Chapter 9. Electrical

conductor coexist in an electrical configuration, the magnetic field that one conductor generates
affects the other, meaning that there will be a geometrical relationship affecting the inductance
of the overhead line.

Equations (9.29) and (9.30) refer to the calculation of the inductance for simplex and duplex
con- figurations respectively.
𝜇0
𝐿 = 2𝜋 · ln 𝐺 · 𝑙 (9.29)
𝐺𝑀𝑅 !
𝜇0
𝐿=
· ln √ 𝐺𝑀√ ·𝑙 (9.30)
𝐷 ︁
2𝜋 𝐺𝑀𝑅 𝐺𝑀𝐷𝑝𝑝
·
Where:

• 𝐿 is the inductance of a fully transposed overhead line [H]

• 𝜇0 is the vacuum permeability [H/m]
• 𝐺𝑀𝐷 is the Geometrical Mean Distance calculated with Equation (9.16) [m]
• 𝐺𝑀𝑅 is the Geometrical Mean Radius calculated with Equation (9.18) [m]
• 𝐺𝑀𝐷𝑝𝑝 is the Geometrical Mean Distance of same phases in a duplex configuration
calcu- lated with Equation (9.17) [m]
• 𝑙 is the length of the line [𝑚]

In order to understand Equation (9.30), it is important to take into account that each circuit’s
phase will have a counterpart in the other circuit. For the calculation model, both phases’ (a and
a’) voltage and current will be considered to have the same argument and power factor.
Knowing this, they are considered to be the same phase when calculating the GMR. If this was
the case, the GMR according to Equation (9.18) would be calculated as:
,v 𝑛 𝑛
v,
𝑛 v𝑛
. . .
𝑑 1→𝑖 ·
2𝑛
𝑑 1→𝑖 𝑑 1→𝑖 ′ (9.31)
𝐺𝑀𝑅 = 2 𝑛
′ 𝑑 ′= 2 𝑛 · ,′
2𝑐𝑖𝑟𝑐𝑢𝑖𝑡𝑠 𝑖=2 𝑖 =1 1→𝑖 𝑖=2 𝑖 =1
′
′
𝑟 · 𝑟 · .

This expression can be simplified using Equations (9.18) and (9.14):

√ √︁
𝐺𝑀𝑅2𝑐𝑖𝑟𝑐𝑢𝑖𝑡𝑠 = 𝐺𝑀𝑅 · 𝐺𝑀𝐷𝑝𝑝 (9.32)

Where 𝑑 1→𝑖 is the distance between the first conductor of the first circuit to the conductors in its
bundle and 𝑑 1 𝑖 ′ is the distance between the first conductor of the first circuit and the
→
conductors of the second circuit’s bundle.

Once the inductance of the line has been obtained, the inductive reactance of the line can be
calculated according to Equation (9.33)

Where:
Overhead line 5
 Chapter 9. Electrical

𝑋𝐿 = 2𝜋 ·𝑓 ·𝐿 (9.33)

• 𝑋𝐿 is the reactive inductance per unit length of the fully transposed overhead line [Ω]

Overhead line 5
 Chapter 9. Electrical

• 𝑓 is the frequency of the line [Hz]

• 𝐿 is the inductance of the line

9.2.5 Impedance
The impedance of the transmission line will be calculated using Equation (9.34)

𝑍→ = 𝑅 + 𝑗𝑋𝐿 (9.34)

Where:

• 𝑍→ is the impedance of the line [Ω]

• 𝑅 is the resistance of the line [Ω]
• 𝑋𝐿 is the inductive reactance of the line [Ω]

9.2.6 Capacitance and Susceptance

The capacitance of an overhead line can be calculated using Equations (9.35) and (9.36) :

2𝜋𝜖0
𝐶= · 𝑙≈ 𝜋𝜖0 · 𝑙
2𝐺𝑀𝐷 (9.35)
l 𝐺𝑀𝐷
𝐺𝑀𝑅 √ �
𝐺𝑀
1+

𝐶=
𝜋𝜖0
2𝐺𝑀𝐷 ·𝑙 (9.36)
𝑙𝑛 𝐺𝑀𝑅𝐶 ·𝐺𝑀𝐷𝑝𝑝

Where:

• 𝐶 is the capacitance of the overhead line [F]

• 𝜖0 is the dielectric constant [F/m]
• 𝐺𝑀𝐷 is the geometrical mean distance according to Equations (9.9) and (9.13) [m]
• 𝐺𝑀𝑅𝐶 is the geometrical mean radius for the capacitance according to Equation (9.19) [m]
• 𝐺𝑀𝐷𝑝𝑝 is the Geometrical Mean Distance of same phases in a duplex configuration
calcu- lated with Equation (9.17) [m]
• hM is the average height of the conductor throughout the overhead line, calculated for
every span. This is obtained by averaging the vertical distance between the catenary curve
(see Section 10.3 for more details) and the terrain throughout the overhead line. [m]
• l is the length of the line [𝑚]

Equation (9.35) shows a precise and a simplified version for the calculation of the capacitance
in a simplex configuration. For conductors with considerable height above ground, both
equations will yield similar results, but for lower conductors, the differences might go up to a 5
to 10% . Therefore, the complete Equation (9.35) has been selected for simplex configurations.

Regarding duplex configuration, the simplified version in Equation (9.36) has been used due to

Overhead line 5
 Chapter 9. Electrical

a slightly higher complexity in its calculation.

Overhead line 5
 Chapter 9. Electrical

Once the capacitance has been calculated, the capacitive reactance and susceptance can be ob-
tained using Equations (9.37) and (9.38)

𝑋𝐶 = 1 (9.37)
2𝜋 · 𝑓 · 𝐶

𝐵 = 2𝜋 · 𝑓 · 𝐶 (9.38)

Where:

• 𝑋𝐶 is the capacitive reactance [Ω]

• 𝑓 is the frequency of the line [Hz]
• 𝐶 is the capacitance per unit length [F/m]
• 𝐵 is the susceptance per unit length [Ω-1 m-1]

9.3 Voltage drop

The voltage drop across an overhead line is defined as the difference between the voltage at the
receiving and the voltage at the sending end. This is an important value that will determine
the quality of the transmission, and will be limited to a 5% maximum voltage drop. The general
approach in the literature is to respect this 5% and sometimes even a 7.5% or a 10% for extreme
designs.

From the distributed parameters model in Subsection 9.1.1, the voltages and currents at the
send- ing and receiving end can be obtained. The power factor at the sending end will be
considered to be equal to the one established by the user in the interface. Therefore, the voltage
and current at the sending end can be determined as:

𝑈→ 𝑆
=𝑈 (9.39) 𝐼→ = (cos 𝜙 + 𝑗 sin 𝜙 ) (9.40)
𝑠 𝑠 √ 𝑠 𝑠
3𝑈
Where:

• 𝑈𝑠 is the phase to phase voltage at the sending end [V]

• 𝑈 is the phase to phase voltage level of the overhead line [V]
• 𝐼𝑠 is the current at the sending end [A]
• 𝑆 is the apparent power of the line [MVA]
• 𝜙𝑠 is the power factor angle at the sending end [rad]

Having the electrical parameters calculated, the voltage and current at the receiving end can be
calculated using Equations (9.7) and (9.8) respectively. The voltage drop can then be obtained
using Equations (9.41) and (9.42) , where ∆V is the voltage drop, Us is the voltage at the
sending end and Ur is the voltage at the receiving end.

Δ𝑈 = |𝑈𝑠 | − |𝑈𝑟 | (9.41)

Overhead line 5
 Chapter 9. Electrical

|𝑈𝑠 | − |𝑈𝑟 |
Δ𝑈 % = 100 (9.42)
( ) |𝑈𝑠 |

9.4 Power factor

The power factor calculation of grid connected transmission lines is a complex problem because
it involves different generation facilities connected simultaneously affecting the power factor of
the line. Because information about other facilities is not available, the power factor calculation
is based on the fact that the current and voltage at the substation are known, according to
Equations (9.39) and (9.40).

Therefore, the voltage and current at the receiving end can be calculated with the distributed
parameters model, giving out a complex value for the voltage and for the current at the
receiving end. The phase difference between these two values will determine the power factor at
the receiving end:

cos 𝜙𝑟 = cos 𝑎𝑟𝑔𝑈→𝑟 − 𝑎𝑟𝑔𝐼→𝑟 (9.43)

Where:

• cos 𝜙𝑟 is the power factor at the receiving end

• 𝑈→𝑟 is the voltage at the receiving end [V]
• 𝐼→𝑟 is the current at the receiving end [I]

9.5 Losses
The purpose of an overhead line is to transmit energy from a generation point to a delivery
point. The power losses during the transmission and distribution will reduce the energy
delivered with the same proportion. This means that, if 5% of the power is lost in the overhead
line, then 5% of the energy carried will also be lost. Because of this, it is really important to
reduce the power losses as much as possible.

The electrical losses in an overhead line are generally caused by two physical phenomena: the
Joule effect, due to the conductor’s resistance and the Corona effect, caused by the ionization of
the air around the conductor. The accepted losses for an overhead line will always be lower than
5% combining the Joule effect and the corona effect.

9.5.1 Joule effect

The well known Joule effect losses appear when a current flows through a conductor with a re-
sistance, dissipating energy as heat. The voltage and current of both, sending and receiving ends
can be calculated using the Equations (9.7) and (9.8) from the hyperbolic distributed parameters
model. The Joule losses can then be calculated according to Equations (9.44), (9.45), (9.46) and
(9.47).

√
𝑃𝑠 = 𝑈𝑠 · 𝐼𝑠 · 3 · cos 𝜙𝑠 (9.44)

Overhead line 6
 Chapter 9. Electrical

√
𝑃𝑟 = 𝑈𝑟 · 𝐼𝑟 · 3 · cos 𝜙𝑟 (9.45)

Δ𝑃 = 𝑃𝑟 − 𝑃𝑠 (9.46)

𝑃𝑟 − 𝑃𝑠
Δ𝑃 % = 100 (9.47)
( ) 𝑃𝑠
Where:

• 𝑃𝑠 is the power at the substation [W]

• 𝑃𝑟 is the power at the end of the line [W]
• Δ𝑃 is the power losses [W]
• 𝑈𝑠 is the voltage at the substation [V]
• 𝑈𝑟 is the voltage at end of the line [V]
• 𝜙𝑠 is the power factor at the substation
• 𝜙𝑟 is the power factor at the end of the line according to Equation (9.43)

9.5.2 Corona losses

Corona effect happens when the air surrounding a conductor gets ionized, becoming conductive
and dissipating some of the overhead line’s energy. This effect usually occurs in high voltage
lines, where the disruptive voltage for air to be ionized is surpassed. The corona effect in an
overhead line is usually calculated according to Peek’s empirical formula [20] , obtaining first
the corona disruptive voltage and then calculating the corona losses if this disruptive voltage is
surpassed in the line.

The corona disruptive voltage is calculated according to Equation (9.48) , which is derived from
Peek’s formulae:

√ 𝐺𝑀
𝑈𝑝 = 3 · 29.8 · 𝑚 · 𝛿 · 𝑚𝑡 𝐷 (9.48)
√ �
·𝑟 ·𝑛 · l
𝑟
Where:

• 𝑈𝑝 is the critical disruptive phase to phase voltage [kV]

• 𝑚𝑐 is the rugosity coefficient of the conductor considered as 0.85 for stranded conductors
• 𝛿 is the air correction factor, calculated according to [3] with T being the temperature
and H being the altitude. It is important to note that Peek establishes the unity of the air
correction factor at standard conditions (25ºC and 1 atm) instead of normal conditions

273 + 25
𝛿= � · exp (−0.00012 · 𝐻 ) (9.49)

• 𝑚𝑡 is the weather correction factor, considered as 0.8 for rainy conditions

• 𝑟 is the conductor’s radius [cm]

Overhead line 6
 Chapter 9. Electrical

• 𝑛 is the number of conductors in the bundle

• 𝐺𝑀𝐷 is the geometrical mean distance according to Equation (9.16) [cm]

If the maximum voltage of the overhead line surpasses the disruptive voltage, the Corona effect
will occur. To calculate the losses, Peek’s equation will be used:

241 √ 𝑟 −5
𝑃𝐶𝑜𝑟𝑜𝑛𝑎 = ( 𝑓 + 2 ) ︂ 𝐺𝑀 𝑈𝑚𝑎𝑥√− 𝑈𝑝 2
· 10 (9.50)
𝛿
𝐷
Where:

• 𝑃𝐶𝑜𝑟𝑜𝑛𝑎 is the power losses due to corona effect [W]

• 𝛿 is the air correction factor
• 𝑓 is the frequency [Hz]
• 𝑟 is the radius of the conductor [m]
• 𝐺𝑀𝐷 is the geometrical mean distance of the line [m]
• 𝑈𝑚𝑎𝑥 is the maximum voltage of the line [kV]
• 𝑈𝑝 is the critical disruptive voltage between phase and earth according to Equation (9.48)
[kV]

As a design requirement, the combination of both, Joule and Corona losses, cannot be higher
than 5% . If this percentage is surpassed, the electrical configuration must be changed. One way
to drastically reduce the effect of corona is by increasing the number of conductors per bundle,
which will increase the disruptive voltage.

Overhead line 6
 Chapter 10. Mechanical

Chapter 10

Mechanical calculation

The mechanical calculation of an overhead line is an essential step in the design, ensuring that
none of the elements will fail during almost every condition. An overhead line is a set of towers,
conductors, insulators, earth wires, fittings and foundations, all of them subject to forces and
weather conditions.

In this chapter, the mechanical calculation of the towers, conductors, earth wires and insulators
will be explained, along with the design requirements for each one of them to prevent failure of
the overhead line. One extra consideration is the strength coordination of the different elements
to avoid that a failure in any component leads into a failure of the entire line.

All of the calculations that apply to conductors in this chapter will be also applicable to the earth
wires.

10.1 Conductor Loads hypothesis

The elements of the overhead line are exposed to the different weather conditions, such as ice,
wind and temperature. Although these conditions mostly depend on the location of the line,
there will only be five hypotheses that will be addressed in this section according to IEC 60826
[18] and CIGRE Technical Brochure 273 [21]:

• Wind: When the elements of the overhead line are exposed to high wind speeds,
there will be a transversal force in the conductors, which will be transferred to the
towers and the insulators. The determination of the wind speed can usually be done from
a statistical analysis of wind speed in the area, however, a wind speed of 33.3 m/s or
120 km/h has been selected, as it is a widely used value in some countries. The
conductor’s temperature used for this hypothesis is 10ºC. This is because IEC [18]
establishes that maximum wind under average of the daily minimum temperatures
(assumed as 10ºC) condition is the most restrictive one regarding wind.
• Ice: The second hypothesis considers the ice accretion in the conductors, which leads into
a noticeable increase in their weight. The calculation of ice loads considers an ice
thickness of 30 mm and an ice density of 900 kg/m 3 , with a conductor’s temperature of -
5ºC and without wind.
• Heavy load: The third hypothesis combines the effect of wind and ice simultaneously,
yielding an increase in both vertical and transversal loads. Again, every country has a

Overhead line 6
 Chapter 10. Mechanical

specific way to calculate the combined wind and ice loads, but the general consensus is
that the wind load under combined effects is a fraction of the total wind load of the first
hypothesis. Therefore the conditions for combined loads are conductors with full ice load
and 70% wind speed on them, with a conductor’s temperature of -5ºC (IEC establishes
that the least probable wind speed is between a 60% and a 85%).
• EDS: Stands for everyday stress, and refers to calm weather conditions, with a
temperature of 15ºC and no wind or ice loads. Under EDS conditions, the objective is to
study the fatigue caused by the effect of aeolian vibrations [18] [21] . Assuming that the
conductors are equipped with vibration dampers, the allowable tensile strength of the
conductors will be considered to be equal to 22% of the conductor’s tensile strength (IEC
specifies a range of values between 15% and 25% , but a 22% has been used to match the
Spanish standard).
• Maximum Temperature: Under this hypothesis, the conductors are under maximum
temperature conditions of 85ºC. Although the ambient temperature will never reach these
values, the power carried by the overhead line will increase it, reaching higher
temperature values. This hypothesis is mainly used to study the sagging of the conductor.

10.2 Loads calculation

In this section, the calculation of the loads per unit length for the aforementioned hypotheses is
presented. These loads have to be calculated for the conductors, the earth wires and the insula-
tors; hence all of them must be able withstand the required loading conditions.

10.2.1 Weight loads

When a conductor is placed in an overhead line, its weight will generate a vertical load at
the attachment points of the towers. This load will depend in the mass per unit length of the
con- ductor.

The equation that is used to calculate the weight loads of a conductor or earthing wire is the
following:

𝑄 𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑚𝑐 · 𝑔 (10.1)

Where:

• 𝑄 𝑤𝑒𝑖𝑔ℎ𝑡 is the weight load [N/m]

• 𝑚𝑐 is the mass per unit length of the conductor [kg/m]
• 𝑔 is the gravity, with a value of 9.81 [m/s2]

10.2.2 Wind loads

When there is wind in an overhead line, it produces a force in the components of the line in the
direction of the wind. As stated previously, the wind estimation can be obtained from meteo-
rological measurements, considering a certain return period (see IEC 60826, section 6 for more
information [18] ). When this information is not available, a common approach is to use a spe-
cific value of wind speed that will ensure the safety requirements of the overhead line, in this
case 120 km/h.

Overhead line 6
 Chapter 10. Mechanical

From the wind speed, the wind pressure can be obtained from Equation (10.2) extracted from
IEC:

1
𝑞0 = · 𝜏 · 𝜇 (𝑉𝑅)2 (10.2)
2

Where:

• 𝑞0 is the wind pressure [Pa]

• 𝜏 is the air density correction factor, considered as 1 (see table 6 of [18] for more details)
• 𝜇 is the air mass per unit volume, which is equal to 1.225 [kg/m3]
• 𝑉𝑅 is the wind velocity, which is considered to be in terrain category B [m/s]

Having the wind pressure from the previous equation, the wind load in the conductor can be
calculated as:

𝑄 𝑤𝑖𝑛𝑑 = 𝑞 0 · 𝐶𝑥𝑐 · 𝐺𝑐 · 𝐺 𝐿 · 𝑑 · sin2 Ω (10.3)

Where:

• 𝑄 𝑤𝑖𝑛𝑑 is the wind load per unit length [N/m]

• 𝐶𝑥𝑐 is the drag coefficient, which is considered to be equal to 1
• 𝐺𝑐 is the combined wind factor for conductors, which is not considered because it rep-
resents the loads of all three conductors in the tower (Equation (10.3) refers to a single
conductor)
• 𝐺𝐿 is the span factor, which has been considered equal to 1
• d is the diameter of the conductor [m]
• Ω is the angle between the wind direction and the conductor, which is considered to be
equal to 90º, meaning that the wind will be perpendicular to the conductor

The final equation that is used to calculate the wind loads per unit length in conductors is the
following:

𝑄 𝑤𝑖𝑛𝑑 = 𝑞 0 · 𝑑 (10.4)

The effect of the wind also generates loads in the overhead line’s insulators, with a leaser effect
in the general design. The expression that is used to calculate the wind loads acting on insulator
strings is:

𝑄 𝑤𝑖𝑛𝑑,𝑖𝑛𝑠 = 𝑞 0 · 𝐶𝑥𝑖 · 𝐺𝑡 · 𝑆𝑖 (10.5)

Where:

• 𝑄 𝑤𝑖𝑛𝑑,𝑖𝑛𝑠 is the wind load at the insulator [N]

Overhead line 6
 Chapter 10. Mechanical

• 𝐶𝑥𝑖 is the drag coefficient of insulators, which is equal to 1.2

• 𝐺𝑡 is the combined wind factor, which is also disregarded since it takes into account the
effect of the wind load for the three insulator strings of the tower.
• 𝑆𝑖 is the area of the insulator considering it to be a rectangle. It can be calculated by
multiplying the length of the insulator and the diameter of one of its elements. [m2]

10.2.3 Ice loads

The ice loads in an overhead line consists of frozen water that adheres to the different elements,
increasing the weight that these elements have to withstand. There are two different types of
ice: precipitation ice and cloud ice. Although the density of these two types of ice is different,
and some designers might take this into account, this calculation model will only consider ice
with a density of 900 kg/m3 .

According to IEC 60826 [18], the ice loads per unit length can be calculated as:

−
𝑄𝑖𝑐𝑒 = 9.82 · 10 3 · 𝛿 · 𝜋 · 𝑡 · (𝑑 + 𝑡 /1000) (10.6)

Where:

• 𝑄𝑖𝑐𝑒 is the ice loads per unit length [N/m]

• 𝛿 is the ice density, which is considered to be 900 [kg/m3]
• 𝑡 is the radial ice thickness, which is considered to be 30 [mm]
• 𝑑 is the conductor diameter [m]

It is important to consider that the calculation of ice loads differs a lot depending on the
country’s standard. Because of this discrepancy, an ice thickness of 30 mm was selected, in
order to be more conservative regarding the calculation, but trying to respect most standards.

In Table 10.1 for the same 30 mm diameter conductor, the ice loads per unit length might go
from 6.5 N/m in Greece up to 79 N/m in Ireland. In order to have a conservative approach, the
conditions selected for the ice loads are an ice thickness of 30 mm and an ice density of 900
kg/m3.

Table 10.1: Ice loads for different countries for a conductor with a 30 mm diameter. [Source: [3]
CountryIce thickness [mm]Ice density [kg/m3 ]Ice load [N/m]
Belgium 20 600 19
Germany Undefined Undefined 8 - 32
Spain Undefined 750 9.9 - 29.6
]
France 20 600 19
Greece 6.35 or 12.7 900 6.5 or 15.3
Ireland 40 900 79
Italy 12 920 19

The ice loads in the insulator strings are not calculated due to the small effect that they have in
the calculation model.

Overhead line 6
 Chapter 10. Mechanical

10.2.4 Total loads

To take into consideration the different weather conditions, the combination of the weight of
the conductor, the wind loads and the ice loads must be calculated, taking into account their
direction. The ice and weight loads will be considered to be vertical loads, whereas the wind
loads will be considered perpendicular to the conductor’s trajectory.

The total loads per unit length are calculated according to:

√︃
2 (10.7)
𝑄 = 𝑄 𝑤𝑒𝑖𝑔ℎ𝑡 + 𝑄𝑖𝑐𝑒 + 𝑄2 𝑤𝑖

Where:

• 𝑄 is the total loads per unit length of the conductor [N/m]

• 𝑄 𝑤𝑒𝑖𝑔ℎ𝑡 is the weight loads per unit length of the conductor obtained from Equation (10.1)
[N/m]
• 𝑄𝑖𝑐𝑒 is the ice loads per unit length of the conductor according to Equation (10.6) [N/m]
• 𝑄 𝑤𝑖𝑛𝑑 is the wind loads per unit length of the conductor according to Equation (10.3) [N/m]

10.3 Catenary
10.3.1 Catenary curve calculation
When a conductor is placed between two attachment points with a tensile strength, it will natu-
rally sag, following a curve called catenary, which is formed as a result of the vertical weight of
the conductor along the span.

It is important therefore to calculate the catenary curve in order to ensure that the
clearances and the tensile strengths are respected according to the IEC requirements. In
order to calculate the catenary, two models are typically used in projects: the hyperbolic model
and the parabolic model.

In this section, the hyperbolic model will be used, as it is more accurate than the parabolic one,
but with higher complexity. In Figure 10.1, the catenary curve is shown with some of its
characteristic points. All the equations will be referred to the coordinate system showed,
considering the origin x=0 at the lowest point of the curve.

Overhead line 6
 Chapter 10. Mechanical

Figure 10.1: Catenary of a conductor

It is also important to note that this section will not study deeply the catenary equations. For
more detailed calculations, check Chapter 14 of [3]

The main equation that describes the catenary curve according to the hyperbolic model is the
following:

𝐻 𝑤𝑐 · 𝑥
𝑦= cosh (10.8)
𝑤𝑐 · 𝐻

Where:

• 𝑦 is the vertical coordinate of the catenary [m]

• 𝐻 is the horizontal tension of the conductor [N]
• 𝑤𝑐 is the weight per unit length of the conductor plus the ice if there was ice accretion
[N/m]
• 𝑥 is the horizontal coordinate of the catenary [m]

The previous equation can be expressed in a more compact way with the definition of the cate-
nary constant c:
𝐻
𝑐= (10.9)
𝑤𝑐

𝑥
𝑦 = 𝑐 cosh (10.10)
· 𝑐

The length of the conductor can be calculated from Equation (10.11) , with a being the span
length in meters and h being the altitude difference between the attachment points in meters.

√︂
. .
𝑎 2
𝐿= ℎ2 + 2𝑐 · sinh 2 (10.11)

In order to calculate the distance from the lowest point of the curve x=0 (see Figure 10.1) to
both attachment points, equations (10.12) and (10.13) are used.

Overhead line 6
 Chapter 10. Mechanical

𝑐
𝑥𝐴 = 𝑐 · 𝑙𝑛 · (1 − exp (−𝑎/𝑐)) (10.12)
(𝐿

𝑥𝐵 = 𝑥𝐴 + 𝑎 (10.13)

Where:

• 𝑥𝐴 is the horizontal distance from the lowest point of the catenary to the first attachment
point [m]
• 𝑐 is the catenary constant [m]
• 𝐿 is the length of the cable [m]
• ℎ is the height difference between the two attachment points [m]
• 𝑎 is the span length [m]
• 𝑥𝐵 is the horizontal distance from the lowest point of the catenary to the second
attachment point [m]

With the previous values of the catenary, the point of maximum sag can be calculated [22] using
Equation (10.14) with h being the height difference, a being the span length and xmin being the
point of lowest altitude, which is equal to 0.

𝑥𝑐 ℎ
= 𝑥𝑚𝑖 + 𝑐 · 𝑎𝑟𝑠ℎ (10.14)

The sag at any point of the catenary can be calculated using Equation :

ℎ · (𝑥 − 𝑥𝐴) 𝑥𝐴 𝑥
𝑓 = � + 𝑐 · cosh � − cosh � (10.15)

Where:

• 𝑓 is the sag at point x [m]

• ℎ is the difference in height between the two attachment points [m]
• 𝑥 is the horizontal distance from the lowest point of the catenary to the point of
interest [m]
• 𝑥𝐴 is the horizontal distance from the lowest point of the catenary to the first attachment
point, calculated from Equation (10.12) [m]
• 𝑎 is the span length [m]
• 𝑐 is the catenary constant calculated from Equation (10.9) [m]

10.3.2 Horizontal tension

The most important value that defines the mechanical calculation of a conductor is the
horizontal tension, which is the horizontal component of the conductor tensile force. This
value plays an important role in defining the catenary of the conductor. In this regard,
higher horizontal tensions will result in smaller sags, reducing the height requirements for the

Overhead line 6
 Chapter 10. Mechanical

support. However,

Overhead line 7
 Chapter 10. Mechanical

the horizontal tension should be limited to ensure that the rated tensile strength (RTS) of the
conductor is not reached, which would lead into a mechanical failure.

According to table 20 of IEC 60826 [18] , the typical range of tensile strength limit is between a
70 % and an 80 % of the RTS. Values higher than that would result in potential damages in the
conductors, meaning that the target of the design will be to never surpass a 70% of the
conductor’s RTS under any meteorological condition. This requirement also affects the design
of earth wires.

10.3.3 State change equation

The previous equations for the catenary calculation use a specific value of horizontal tension
and mass per unit length of the conductor, but these two values highly depend on the
meteorolog- ical conditions. Because of this, the catenary has to be calculated for the different
hypotheses mentioned in Section 10.2 to ensure a correct design for every condition.

When designing an overhead line for a given hypothesis, the value of the horizontal tension will
be defined by that given hypothesis. However, different conditions will result in different
values, which could potentially not fit the project’s requirements if it was only designed for the
initial hypothesis. Because of this, it is important to study the different cases and design the
project respecting all of them.

The state change equation is used to translate the horizontal tension obtained under one me-
teorological condition into a different one. This is used to check that everything is correctly
sized and that the line is designed according to the most restrictive condition, never surpassing
a certain percentage of horizontal load that could be dangerous to the elements of the line.

The state change equation begins with an initial condition with a specific load per unit length
(Q1), temperature (ϑ1) and horizontal tension (H1) which corresponds to the EDS hypothesis
mentioned in Section 10.1 .
◦
• 𝜃1 = 15 𝐶
• 𝑄 1 = 𝑄 𝑤𝑒𝑖𝑔ℎ𝑡
• 𝐻1 = 0.22 𝑅𝑇 𝑆

From these initial conditions, the horizontal tension H 2 at conditions ϑ2 and Q2 can be obtained
with the state change Equation (10.16)

𝛼 · 𝑆 · (𝑎𝑟 · 𝑄1)2 𝜖 · 𝑆 · 𝛼 · (𝑎𝑟 · 𝑄2)2

𝐻2 − 𝐻1 + 𝜖 · 𝑆 · (𝜃2 − 𝜃1) = (10.16)
𝐻 22 + 24 · 𝐻 2 2
1

Where:

• 𝐻2 is the horizontal tension at final conditions [N]

• 𝐻1 is the horizontal tension at initial conditions [N]
• 𝜖 is the modulus of elasticity of the conductor [Pa]
• 𝛼 is the coefficient of thermal expansion of the conductor [1/ºC]
• 𝑆 is the conductor’s cross section [m2]
• 𝑎𝑟 is the ruling span obtained from Equation (1.1) [m]

Overhead line 7
 Chapter 10. Mechanical

• 𝑄1 is the load per unit length at initial conditions [N/m]

• 𝑄2 is the load per unit length at final conditions [N/m]
• 𝜃1 is the temperature at initial conditions [ºC]
• 𝜃2 is the temperature at final conditions [ºC]

Previous equation can be rewritten in a simplified version in Equation (10.19), which can be
solved using the method proposed in [23]:

𝜖 · 𝑆 · (𝑎𝑟 · 𝑄1)2
𝑎 = −𝐻1 + + 𝜖 · 𝑆 · 𝛼 · (𝜃2 − 𝜃1) (10.17)
24 · 𝐻
2 1

𝛼 · 𝑆 · (𝑎𝑟 · 𝑄2)2
𝑏= (10.18)
2 =𝑏
𝐻 2 [𝐻2 + 𝑎] (10.19)
2
If the horizontal tension H2 gives a value higher than the required 70% RTS, the initial
conditions will be changed so that the requirements for the invalid hypothesis are within the
limits. For next hypotheses, the reference conditions used will correspond to the updated ones.
To better understand this with an example, if the ice hypothesis appears to have a horizontal
tension equal to 80%, the reference conditions would have to be changed to the following:
◦
• 𝜃1 = −5 𝐶
• 𝑄 1 = 𝑄 𝑤𝑒𝑖𝑔ℎ𝑡 + 𝑄𝑖𝑐𝑒
• 𝐻1 = 0.7𝑅𝑇 𝑆

These conditions will be used for the rest of the hypotheses until another one does not respect
the design values.

After ensuring that the horizontal tensions are correct, the different horizontal tensions under
every hypothesis will be calculated using the final state change reference conditions in order to
calculate the catenaries using Equation (10.8).

For the earth wire, the horizontal tension will be such that the sag at EDS conditions is not
higher than 90% of the sag of the conductor at the same conditions [3]. This will ensure that the
clearance between the earth wire and the conductor is respected.

10.4 Tower forces

The forces that the towers of an overhead line withstand are a result of the loads transmitted
by the conductors’ weight and horizontal tension. The towers of the line will have to withstand
vertical, longitudinal and transversal forces for different conditions. This section will present the
different hypotheses for the design of the line as well as the calculation model of the forces. The
calculation model does not dive deep into the detailed forces in the towers and focuses on the
forces at the attachment points of the conductors, not considering the resistances at each part of
the tower (in [3] a more detailed calculation is presented). Additionally, the forces of the wind
and ice in the towers themselves will not be considered, as they account for a small percentage
compared to the ones produced by the conductors.

Overhead line 7
 Chapter 10. Mechanical

10.4.1 Tower load hypothesis

In a similar way as with the conductors and earth wires, the loads on the towers will differ
depending on the meteorological conditions and the status of the overhead line. Because of this,
the following hypotheses have to be studied:

• Wind: The conditions of wind speed and temperature are equal to the ones for the
conduc- tors, with a 120 km/h wind speed and 10ºC ambient temperature. There is no
longitudinal force because the conductors’ horizontal tensions are not unbalanced for
suspension and angle towers.
• Heavy load: The conditions of ice and temperature are equal to the ones used in previous
calculations, with an ice density of 900 kg/m3 , an ice thickness of 30 mm and a
temperature of -5ºC. Additionally, a 70% of wind speed will be considered, and similar to
the previous hypothesis, no longitudinal force will be applied for suspension and angle
towers.
• Unbalance: While being under ice load, there is an unbalance in the horizontal tensions
of the conductors, resulting in a longitudinal force. A fraction of the horizontal tension
under ice conditions will be considered as the longitudinal force depending on the type of
tower: For suspension towers 0.15 multiplied by the horizontal tension; for angle towers
0.5; for dead end towers it is not considered because previous hypothesis already studied
it.
• Conductor break: If a conductor breaks, a severe longitudinal force will appear at the
tower’s attachment point. This hypothesis considers a line with ice and the following lon-
gitudinal forces as a fraction of the conductor’s horizontal tension: For suspension towers
0.5 multiplied by the horizontal tension; for angle towers 1; for dead end towers 1.

10.4.2 Vertical forces

When a tower supports a set of conductors and insulators, these elements have a weight that
generates a vertical force at the different attachment points of the tower. The calculation of the
vertical force can be obtained as a result of adding the conductor’s and insulator’s weight, as
stated in Equation (10.20):

𝐹𝑉 = 𝑚𝑐 · 𝑔 · 𝑛𝑐𝑜𝑛𝑑 · 𝑎 𝑤𝑒𝑖𝑔ℎ𝑡 + 𝑛𝑖𝑛𝑠 · 𝑚𝑖𝑛𝑠 · 𝑔 (10.20)

Where:

• 𝐹𝑉 is the vertical force of one attachment point [N]

• 𝑚𝑐 is the mass per unit length of the conductor [kg/m]
• 𝑛𝑐𝑜𝑛𝑑 is the number of conductors per bundle
• 𝑎 𝑤𝑒𝑖𝑔ℎ𝑡 is the weight span of the tower [m]
• 𝑛𝑖𝑛𝑠 is the number of insulators per phase (one string for suspension and dead end towers
and two strings for angle towers)
• 𝑚𝑖𝑛𝑠 is the mass of an insulator [kg]

The vertical force on the entire tower can be calculated as the force of one attachment point
multiplied by the number of attachment points of the tower (3 per circuit).

Overhead line 7
 Chapter 10. Mechanical

10.4.3 Transversal forces

The forces that are applied to the tower sideways are called transversal forces. They are caused
be both the wind and the deflection of the conductors. Figure 10.2 shows the transversal forces
caused by the conductors in a tower’s attachment point.

Figure 10.2: Transversal forces of a deflected tower with wind

The force caused by the deflection angle will be defined as the angle resultant, and can be calcu-
lated according to Equation (10.21)

𝐹𝛼 = (𝐻𝑇 1 + 𝐻𝑇 2) · sin 𝛼 (10.21)

Where:

• 𝐹𝛼 is the angle resultant [N]

• 𝛼 is the deflection angle [rad]
• 𝐻𝑇 1 is the horizontal tension of the previous span [N]
• 𝐻𝑇 2 is the horizontal tension of the next span [N]

Whenever there is wind present, the force that it produces in the conductors will translate into
a transversal force in the tower, which can be calculated using Equation (10.22)

𝐹 𝑤𝑖𝑛𝑑 = 𝑄 𝑤𝑖𝑛𝑑 · 𝑎 𝑤𝑖𝑛𝑑 · cos 𝛼 · 𝑛𝑐𝑜𝑛𝑑 (10.22)

Where:

• 𝐹 𝑤𝑖𝑛𝑑 is the transversal force caused by the wind on a conductor bundle [N]
• 𝑄 𝑤𝑖𝑛𝑑 is the wind load per unit length of the conductor [N/m]
• 𝑎 𝑤𝑖𝑛𝑑 is the wind span of the tower [m]
• 𝛼 is the deflection angle [rad]
• 𝑛𝑐𝑜𝑛𝑑 is the number of conductors in the bundle

The total transversal force at the tower will therefore be calculated according to Equation (10.23):

Overhead line 7
 Chapter 10. Mechanical

𝐹𝑇 = 𝐹𝛼 + 𝐹 𝑤𝑖𝑛𝑑 (10.23)

The forces calculated with Equations (10.21) and (10.22) define the transversal forces at one at-
tachment point of the tower, consequently in order to calculate the entire transversal forces of
the tower, the values must be multiplied by the number of phases of the tower.

10.4.4 Longitudinal forces

The calculation of the longitudinal forces depends primarily on the percentage of the horizontal
tension that is considered for the different hypotheses. Additionally, the deflection angle of the
tower also defines the longitudinal loads that the conductors tension generates.

The expression used to calculate the longitudinal forces of a conductor bundle is the following:

𝐹𝐿 = 𝛿 · 𝐻 · 𝑛𝑐𝑜𝑛𝑑 · cos 𝛼 (10.24)

Where:

• 𝐹𝐿 is the longitudinal force of a conductor bundle [N]

• 𝛿 is the fraction of the horizontal tension that will be considered as longitudinal force
• 𝐻 is the maximum horizontal tension between the previous and next span’s conductor [N]
• 𝑛𝑐𝑜𝑛𝑑 is the number of conductors in the bundle
• 𝛼 is the deflection angle [rad]

Similarly to the vertical and transversal calculation, the longitudinal forces refer to a single
phase. However, in this case, some of the hypotheses might not take into account the
longitudinal force of all of the conductors, therefore this force is only multiplied by the number
of phases for dead end towers.

Overhead line 7
 Chapter 11. Tower

Chapter 11

Tower spotting

Another important parameter in the investment of an overhead line is the placement of the
towers within the line. Spotting the towers in a way or another impacts directly the cost of
the line. By way of explanation, tower spotting must consider the clearances, the catenary and
all the other elements that define the line.

In this chapter the tower’s spotting criteria and process will be explained.

11.1 Considerations
In this methodology, a line with the least number of angle towers is prioritized, according to [ 3]
an overhead line route which avoids angle towers is favored to be selected. Hence, the angle
towers uploaded by the users are preserved and no other angle tower is added in the spotting
process. If a user imports close angle towers that do not respect the minimum allowable span,
the towers are kept and a warning is displayed in the results.

Furthermore, a target span calculated based on the line voltage is considered to ensure a
tolerated span length, the target span is calculated as seen in Table 2.1. Moreover, the elevation
data under the entire line is considered for placing new suspension towers within the blocks; the
terrain data consideration is necessary to ensure the minimum clearance to the ground criteria.

Before jumping into the spotting process, the line imported by the users goes through a
filter based on the angle towers criteria mentioned above. The angle towers are considered
◦ ◦
when there is a deflection angle higher than 2 or less than -2 ; these towers are kept and
the rest of points/towers are removed.

11.2 Spotting process

The spotting process in this methodology follows three different steps; the first step is called a
distance check where the algorithm ensures that the users blocks respect the allowable distances
and distributes suspension towers depending on the target spans. In the second step, the ground
clearance distance in each span is validated and new possible towers based on the elevation data
are considered. Finally, in the last step, the new towers are spotted with a final check.

Overhead line 7
 Chapter 11. Tower

This process is repeated in a way that the final route respects the minimum ground clearance
and the span length while including the necessary towers only to evacuate the power.

11.2.1 Distance check

In the distance check step, the blocks lengths are evaluated to make sure there is enough space
between all the angle towers; this evaluation considers a tolerance span of 40% of the target
span which a block’s length can tolerate.

In addition, in this step, the suspension towers are located within the blocks based on a maxi-
mum span calculated from the target span defined by the system voltage. The maximum span is
calculated as the target span multiplied by 1.15 to allow up to 15% bigger spans. For every
block imported by the user, its corresponding elevation data is calculated; hence, for every
maximum span the corresponding point is retrieved and considered as a new possible tower.

Later, the possible new towers are added to the line in a way that guarantees all the distances
between the towers respect the target spans, if not, a consideration of new possible towers is
made. Also, new elevation data is defined based on the new spans of the line.

In the Figure 11.1 an illustration of a plan view of a line imported by the user is presented; the
distance check criteria are also shown.

Figure 11.1: The distance check: (a) User line path | (b) Path filtering | (c) Possible towers location
| (d) Suspension towers are added

11.2.2 Ground clearance validation

In order to have a valid path one of the most important criteria is for the towers to be placed
in such a way that the spans’ catenaries respect the minimum ground clearance. To guarantee
this criteria, a ground clearance validation is run on the line composed of the towers from the
distance check step.

Overhead line 7
 Chapter 11. Tower

The process of ground clearance validation includes the calculation of the catenaries of the line
considered from the previous step; afterwards, the maximum height of the tower’ body is calcu-
lated based on the voltage of the system as presented in Table 11.1 which was elaborated based
on market towers catalog [19]. The maximum height is used to define the lowest connection
point of the towers depending on their type to recalculate the correct catenary of the
conductors.

For every two consecutive spans, the minimum ground clearance is validated against the spans’
elevation data. For every span where the ground clearance is not respected, its corresponding
towers are defined as invalid, the ground clearance validation procedure allows keeping track of
all the spans and towers where the clearance is not respected for it to be used to find possible
valid towers in the following step.

Table 11.1: The maximum tower height. Source: Imedexsa [19]

Maximum voltageMaximum tower
Tower name
𝑈 [kV] height [m]
⩽66 30.22 Milano
⩽220
⩽132 39.2
33.05 Condor
Halcon Real
>220 40 Gran Condor/Icaro

In the Figure 11.2 below, an illustration of ground clearance validation of a line composed
of three spans is shown, the towers in red are the invalidated towers because of the invalid
first span that do not respect the ground clearance, the other spans are respecting the criteria;
hence, represented by a green color.

Figure 11.2: The ground clearance validation

Overhead line 7
 Chapter 11. Tower

11.2.3 Valid towers spotting

Finally, in this last step of tower spotting, the invalid spans are evaluated depending on the type
of the tower that compose them. The evaluation results in adding new suspension towers and
creating new spans when needed.

In one hand, when the invalid span is made of two angle towers, a new tower is added at the
middle of the span to correct the ground clearance. On the other hand, if the span contains at
least one suspension tower, the adjacent spans must be evaluated. Based on the validity or
invalidity of the adjacent spans, the towers are moved forward or backward to make sure the
spans respect the minimum ground clearance criteria. The movement of the possible valid
towers considers the minimum allowable span required.

The new positions of the moved towers are selected in a way that high ground positions are
prioritized while trying to keep the towers as equally spaced as possible. Once the new towers
positions are defined, the previous step is run again to make sure the correct spans are
respecting the minimum ground clearance.

In the case of not being able to find a tower’s position that would validate a certain span, adding
an extra tower is considered. In the following Figure 11.3 an illustration of a tower path made of
three spans with the first span being invalid.

Figure 11.3: Tower spotting sample

Overhead line 7
 Bibliograp

Bibliography

[1] Grigsby, L.L., The Electric Power Engineering Handbook - Five Volume Set (3rd ed.) Taylor
and Francis, 2012.
[2] Félix Ignacio Pérez Cicala, Mario Bennekers Vallejo and Miguel Ángel Torrero
Rionegro, “Topography analysis methodology (2022 edition),” RatedPower, 2017.
[3] F. Kiessling, P. Nefzger, J. F. Nolasco, and U. Kaintzyk, Overhead Power Lines:
Planning Design Construction. Springer, 2003.
[4] Technical Committee 7, “Overhead electrical conductors – Calculation methods for
stranded bare conductors,” International Electrotechnical Commision, Tech. Rep. IEC
61597, 2012.
[5] Siemens AG, “Power Engineering Guide Edition 8.0,” Siemens, Tech. Rep., 2017.
[6] “Aerial optical cables along electrical power lines,” ZTT, catalog, 2007.
[7] Technical Committee 14, “Power transformers - part 5: Ability to withstand short
circuit,” International Electrotechnical Commision, Tech. Rep. IEC 60076-5:2006, 2006.
[8] Viesgo, “Norma instalaciones de enlace en alta tension: lineas de alta tension (>36 kV) y
subestaciones,” Viesgo, Technical Specification NT-IEAT.01, 2017.
[9] Viesgo, “Proyecto tpo de subestacionesc con aparamenta convencional,” Viesgo,
Technical Specification PT-SECO.01, 2017.
[10] J. P. Fernandez and E. Iraburu, “Especificaciones Particulares. Requisitos Tecnicos de Con-
struccion de Subestaciones conectadas a redes de Alta Tension de Un > 36 kV,” Union
Fenosa Distribucion, Technical Specification IT.07974.ES-DE.NOR, 2017.
[11] Technical Committee 99, “Insulation coordination. Part 2. Application guidelines,” Inter-
national Electrotechnical Commision, Tech. Rep. IEC 60071-2:2018, 2018.
[12] Ignacio Álvarez Iberlucea, Soukayna Jermouni, Miguel Ángel Torrero Rionegro, Félix
Ig- nacio Pérez Cicala and Juan Romero González, “Substation methodology (2020
edition),” RatedPower, 2020.
[13] Technical Committee 36, “Insulators for overhead lines with a nominal voltage above
1000 V - Ceramic or glass insulator units for AC systems - Characteristics of insulator
units of the cap and pin type,” International Electromechanical Commision, Tech. Rep.
IEC 60305:2021, 2021.
[14] Technical Committee 36, “Insulators for overhead lines with a nominal voltage above
1000 V - Ceramic insulators for AC systems - Characteristics of insulator units of the
long rod type,” International Electromechanical Commision, Tech. Rep. IEC 60433:2021,
2021.

Overhead line 8
 Bibliograp

[15] ABB, Switchgear Manual, 11th ed. 2007.

[16] Technical Committee 36, “Selection and dimensioning of high-voltage insulators
intended for use in polluted conditions - Part 1: Definitions, information and general
principles,” International Electrotechnical Commision, Tech. Rep. IEC 60815-1:2008,
2008.
[17] European Committee for Electrotechnical Standardization, “Overhead electrical lines ex-
ceeding AC 1 kV - Part 1: General requirements - Common specifications,” CENELEC,
Tech. Rep. EN 50341-1, 2012.
[18] Technical Committee 11, “Overhead transmission lines - design criteria,” International
Electrotechnical Commission, Tech. Rep. IEC 60826:2017, 2017.
[19] “Catalogo general,” Imedexsa, catalog, 2012.
[20] F.W. Peek, “The law of corona and dielectric strength of air,” 1912.
[21] “Conductor safe design tension with respect to aeolian vibrations,” CIGRE, Tech. Rep.
CI- GRE Report 273, 2005.
[22] A. Hatibovic, “Derivation of equations for conductor and sag curves of an overhead line
based on a given catenary constant,” 2013.
[23] S.Partheepan and D.Sivalingam, “Solution to cubic equation using java programming,”
2020.

Overhead line 8
 Appendix A. Selecting a phase conductor and an

Appendix A

Selecting a phase conductor and

an insulator

In this appendix, the process and calculation of the phase conductor cross-section as well as the
selection of the line insulator will be presented.

The overhead line to be designed is a transmission line evacuating a capacity of 100MW for a
high voltage level of 132kV and frequency of 50Hz. The system is simplex with one circuit and
one conductor per phase.

A.1 Selection of the phase conductor

As explained in Chapter 2, the selection of the conductor depends mainly on the thermal limit
criteria and the mechanical criterion.

The line is of a transmission nature; hence, the conductors to select from are of type ACSR. In
this example, the conductor used for the selection is the "160-A1/S2A" with a total diameter of
0.0177 m and a unitary mass of 0.64459 kg/m.

The thermal limit withstand is checked by calculating the conductor maximum admissible cur-
rent using Equation (2.3) and comparing it with the total current of the line calculated as:

𝑆 6
𝐼𝑡𝑜𝑡 = 100 ·
𝑈𝑚𝑎𝑥 √
= √ ≈ 437𝐴 (A.1)
·𝑛 · 3 132000 · 1 · 3
To calculate the maximum admissible current, first, the DC resistance at maximum temperature
must be calculated:

−8
𝜌 3.35 · 10 −4
𝑅𝑇 = [1 + 𝛼 (𝑇𝑚𝑎𝑥 − 20)] = [1 + (0.00403(80 − 20))] ≈ 2.234 · 10 𝑊 /𝑚 (A.2)
𝑠 1.8622 · 10−4

Next, the heat loss by radiation, the convection loss and the solar heat gain must be calculated
respectively:

Overhead line 8
 Appendix A. Selecting a phase conductor and an

−8
𝑁𝑅 = 𝑘 · 𝜋 · 𝑑 · 𝐾𝑒 (𝑇𝑚𝑎𝑥 4 − 𝑇𝑎𝑚4) = 5.67 · 10 · 𝜋 · 0.0177 · 0.45(3534 − 3184) ≈ 7.52𝑊 /𝑚 (A.3)

𝑁𝐶 = 𝜆 · 𝑁𝑢 · 𝜋 · (𝑇𝑚𝑎𝑥 − 𝑇𝑎𝑚) = 0.02585 · 25.7 · 𝜋 · (353 − 318) ≈ 73.1𝑊 /𝑚 (A.4)

𝑁𝑆 = 𝑌 · 𝑑 · 𝑆𝑖 = 0.0177 · 1045 · 0.8 ≈ 14.8𝑊 /𝑚 (A.5)

Hence, the maximum admissible current of the conductor is:

√︂ √︂
𝑁𝑅 + 𝑁𝐶 − 𝑁𝑆 7.52 + 73.1 − 14.8
𝐼𝑚𝑎𝑥 = ·𝑛 = · 1 ≈ 543A (A.6)
𝑅𝑇 2.234 · 10−4

Since the conductor admissible current is higher than the total current of the line, the conductor
"160-A1/S2A" is electrically eligible.

Next, the mechanical withstand is checked for the conductor using the empirical approach ex-
plained in section 2.2.2. For the high voltage of 132kV, the maximum target span is 200 m (
Table 2.1). Hence, from Table 2.2 it is deducted that the minimum tensile strength is 45000 N .

On the other hand, the electrical eligible conductor has a maximum load of 61340 N. Conse-
quently, the conductor withstands mechanically.

The final step in the conductor selection is checking the voltage gradient criterion, as mentioned
in the Chapter 2 using Equation (A.7) and ensuring that the line’s voltage drop and power losses
are not exceeding 5%. In Appendix B, both, the voltage drop and total losses are calculated for
this line’s design and both do not exceed the 5% criterion. The voltage gradient is calculated as
follows:
𝐸 = 𝐶𝑖 [1 + 2 · (𝑟 /𝑠) (𝑛2 − 1) · 𝑠𝑖𝑛(𝜋 /𝑛2)] √� =
2
� 𝜋𝜖0 · 𝑛2 · 𝑟 3 · 100
−
8.23 · 10 12 (A.7)
2 · 𝜋 · 8.854 · 10−12 · 1 · 0.00885 132000

[1 + 2 · (0.00885/0.4)(1 − 1) · 𝑠𝑖𝑛(𝜋 /1)] √ ≈ 12.7𝑘𝑉 /𝑐𝑚

3 · 104
As the voltage gradient is less than the maximum 17kV/cm, the conductor "160-A1/S2A" complies
with the selection criteria; hence, it is the conductor of the this transmission line.

A.2 Selection of the insulator

To select the correct insulator for suspension and tension following this methodology, electrical
and mechanical criteria are considered.

For the same transmission line with the conductor "160-A1/S2A" having a maximum load of 61340
N, the insulator selection will depend, first, on the electrical withstand of the cap and pin insula-
tor. The insulator "U160BS" is chosen as an example of insulator selection. To assess the electrical
withstand, the length of the insulator is calculated under different conditions to ensure it
respects the electrical clearances and to guarantee the insulator withstands the maximum

Overhead line 8
 Appendix A. Selecting a phase conductor and an

voltage.

Overhead line 8
 Appendix A. Selecting a phase conductor and an

- Normal conditions:

−6
𝑛𝑛𝑜𝑟𝑚𝑎𝑙 ≥ 𝑈𝑠 · 𝜖0 = 145000 · 20 · 10 = 9 21 (A.8)
𝜖 0.

- Wet and lightning impulse conditions:

For both conditions, the number of disks that withstand the maximum wet voltage and the light-
ning impulse voltage is seven.

Hence, for the three conditions the longest insulator is composed of 10 pins, which is the integer
immediately superior to 9.21; and for a phase to earth clearance of 1.2m, the insulator minimum
length is calculated as:

𝐿𝑖𝑛𝑠 = 𝑑𝑝𝑝 · 1.1 = 1.2 · 1.1 = 1.32𝑚 (A.9)

Since the insulator total length 10 0.146 = 1.46 is higher than the minimum length, then the
( ·
insulator is electrically compliant.

Next, the minimum failing load of the insulator must be checked against the conductor
maximum load at worst conditions. The different combinations of insulators sets are also
assessed. As the minimum failing load is 160000 N and higher than the conductor maximum
load, the insulator "U160BS" withstands mechanically.

Finally, as explained in the chapter, the typical tensile strength for insulators under the voltage
of 132kV is 160000 N Table 6.3 which is equal to the insulator "U160BS" minimum failing
load. Hence, the said insulator is selected.

Overhead line 8
 Appendix B. Electrical parameters

Appendix B

Electrical parameters calculation

In this appendix, the process and calculation of the electrical calculations of an overhead line are
presented.

The overhead line in question has the same characteristics mentioned in Appendix A. The line
operating under a voltage of 132kV has a length of 8955m and a power factor at the sending end
of 0.95. As explained in Chapter 9, the distributed parameters model will be considered for the
electrical calculation.

B.1 Resistance
The resistance of the line per unit length is calculated using Equation (9.28) as follows:

𝑅𝐴𝐶 −4
2.24 · 10
𝑅= ·𝐿 = · 8955 ≈ 2Ω/𝑚 (B.1)
𝑛𝑐 · 1·1
𝑛𝑠

B.2 Inductance and reactance

To calculate the inductance of the line, the GMD and GMR must be calculated. An accurate GMD
is calculated using Equation (9.16)

.𝑠𝑝𝑎𝑛𝑠
𝑖=1
𝐺𝑀𝐷 𝑜ℎ𝑙 ≈ 7.9𝑚 (B.2)
= 𝐿𝑜ℎ𝑙

𝐺𝑀
As for GMR, it is calculated using Equation · 𝐿𝑖
(9.18): 𝑖

v,
𝑛
.
𝑑 1→𝑖 ≈ 0.0069𝑚 (B.3)
𝐺𝑀𝑅 = 𝑛
𝑟′· 𝑖=2

Next, as the transmission line is Simplex with one circuit, the inductance of the line is calculated
using the following formula:

Overhead line 8
 Appendix B. Electrical parameters

−6
𝜇0 𝐺𝑀𝐷 1.256 · 10 7.9
𝐿= · ln ·𝑙 = · ln · 8955 ≈ 0.0126𝐻 /𝑚 (B.4)
2𝜋 𝐺𝑀𝑅 2𝜋 0.0069

Finally, the inductive reactance can be deduced from Equation (9.33) as follows:

𝑋𝐿 = 2𝜋 · 𝑓 · 𝐿 = 2𝜋 · 50 · 0.0126 ≈ 3.97Ω (B.5)

B.3 Impedance
Having the resistance and the inductive reactance of the line calculated in the previous sections,
the impedance of the entire line can be calculated using Equation (9.34):

𝑍→ = 𝑅 + 𝑗𝑋𝐿 = 2 + 𝑗 3.97 = 4.44Ω (B.6)

B.4 Capacitance and susceptance

The capacitance of a simplex line can be calculated following the Equation (9.35):

−12
𝐶=
2𝜋𝜖0
·𝑙 = 2𝜋 · 8.854 · 10 · 8955 ≈ 7.37 · 10− 𝐹 / (B.7)
ln √ 𝐺𝑀𝐷
2
ln 8 𝑚
√ 7.9
2
𝐺𝑀𝑅𝐶 · 1+ 0.00885· 1+(7.9/2·14.7)
(𝐺𝑀𝐷/2ℎ𝑀 )

Once the capacitance is calculated, the capacitive reactance and the susceptance are calculated
as follows:

1 1
𝑋𝐶 = = ≈ 43203.7Ω/𝑚 (B.8)
2𝜋 · 𝑓 · 𝐶 2𝜋 · 50 · 7.37 · 10−8

−8 −
𝐵 = 2𝜋 · 𝑓 · 𝐶 = 2𝜋 · 50 · 7.37 · 10 ≈ 2.32 · 10 5 Ω−1𝑚−1 (B.9)

B.5 Voltage drop

To calculate the voltage drop using the distributed parameters electrical model, first, the voltage
at the receiving end must be identified. This latter is calculated using Equation (9.7):

𝑉 =𝑉𝑠 · cosh 𝛾 − 𝐼𝑠 · 𝑍0 · sinh 𝛾 =

−5
(76210.2 + 𝑗 0) · (0.99 + 𝑗 2.32 · 10 ) − (415.5 − 𝑗 136.57)· (B.10)
(426.17 − 𝑗 101.58) · (0.00235 + 𝑗 0.00986) = (74832.5 − 𝑗 1372.03)

Consequently, the voltage drop and its percent are calculated as follows:

Overhead line 8
 Appendix B. Electrical parameters

Δ𝑉 = |𝑉𝑠 | − |𝑉𝑟 | ≈ 2364.5𝑉 (B.11)

Overhead line 8
 Appendix B. Electrical parameters

|𝑉𝑠 | − |𝑉𝑟 | 2364.5

Δ𝑉 (%) = · 100 = · 100 ≈ 1.8% (B.12)
|𝑉𝑠 | 132000

B.6 Joule losses

The joule effect loss is the difference between the power loss at the sending loss and that of the
end of the line. To calculate the power losses, the current at receiving end must be calculated
using Equation (9.8):
−
𝐼 =(415.5 − 𝑗 136.57) · (0.99 + 𝑗 2.32 · 10 5)−
(76210.2 + 𝑗 0) (B.13)
(426.17 − 𝑗 101.58) · (0.00235 + 𝑗 0.00986) = (415.5 − 𝑗 138.3)

Next, with the receiving end power factor being 0.9544, both the power losses at sending and
receiving ends are calculated as follows:

√ √
𝑃𝑠 = 𝑉𝑠 · 𝐼𝑠 · 3 · cos 𝜙𝑠 = 132000 · 437.38 · 3 · 0.95 ≈ 9.5 · 107𝑊 (B.14)

√ √
𝑃𝑟 = 𝑉𝑟 · 𝐼𝑟 · 3 · cos 𝜙𝑟 = 74845.06 · 437.92 · 3 · 0.9544 ≈ 9.4 · 107𝑊 (B.15)

Finally, the joule effect and its percent are calculated using Equation (9.46) and Equation (9.47):

Δ𝑃 = 𝑃𝑟 − 𝑃𝑠 ≈ 1151654𝑊 (B.16)

𝑃𝑟 − 𝑃𝑠
Δ𝑃 % = 100 1.2% (B.17)
( ) 𝑃𝑠 · ≈

B.7 Corona losses

In case of the occurrence of the Corona effect, its corresponding loss is calculated; to know the
if the effect occur, the critical disruptive voltage is calculated first using Equation (9.48):

29.8 𝐺𝑀𝐷
𝑉𝑝 = √ ·0.85·0.96553·0.8·0.885·1·ln ≈ 144139.4𝑉 (B.18)
√ ·𝑚𝑐 ·𝛿 ·𝑚𝑡 ·𝑟 l = 22 0.885
0.008
·𝑛·2 𝑟

As the maximum voltage of the line 145kV is higher than the disruptive voltage, the corona
effect will occur and its loss is calculated as follows:

241 𝑟 −5
𝑃𝐶𝑜𝑟𝑜𝑛𝑎 = · ( 𝑓 + 2) 𝐺𝑀 2 · 10
·𝛿 𝐷 𝑉𝑚𝑎𝑥 − 𝑉𝐶𝑚𝑎𝑥

Overhead line 8
 √︂ Appendix B. Electrical parameters

√
= · (50 + 25) · (B.19)
0.96553 √︂ 0.885 145000 − 144139.4
2 −
792.03 2 √ · 10 5 ≈
0.0016𝑊

Overhead line 9
 Appendix B. Electrical parameters

The corona loss is then multiplied by the total number of phases, the number of sub-conductors
and the length of the line to get the corona loss of the line resulting in this case a loss of 41.5W.

Finally, the total losses percentage is:

Δ𝑃 + 𝑃𝐶𝑜𝑟𝑜𝑛𝑎 1151654 + 41.5

𝑃 𝑡𝑜𝑡 = 100 = 100 ≈ 1.15% (B.20)
� 1000

Overhead line 9
 Appendix C. Mechanical

Appendix C

Mechanical calculations

This appendix presents the different mechanical calculations performed for the design of an
over- head line. The appendix will cover the conductor’s loads and the catenary of a span under
dif- ferent hypotheses as well as the tower forces of a suspension and a tension tower.

C.1 Calculation of the loads

For the different hypotheses mentioned in Section 10.2, the loads per unit length of the
conductor can be calculated. The conductor’s characteristics necessary for the mechanical
calculation are presented in Table C.1:

Table C.1: Mechanical data of the conductor "160-A1/S2A"

DiameterUnitary MassTensile Section [m2]ElasticityThermal [Pa]Expansion [1/ºC]
[m][kg/m]Strength [N]
0.0177 0.6449 61340 1.862·10-4 7.454·1010 1.887·10-5

The weight load per unit length can be calculated from (10.1):

𝑄 𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑚𝑐 · 𝑔 = 0.6449 · 9.81 = 6.326 𝑁 /𝑚 (C.1)

The wind pressure is obtained from (10.2):

1 1
𝑞𝑚𝑎𝑥 = 𝜏 𝜇 (𝑉𝑚𝑎𝑥 ) 2 = · 1 · 1.225 · 33.332 = 680.42 𝑁 /𝑚 2 (C.2)
2 2

1 1
𝑞75% = 𝜏 𝜇 (𝑉75%)2 = · 1 · 1.225 ∗ (0.75 ∗ 33.33)2 = 382.74 𝑁 /𝑚2 (C.3)
2 2

The wind load per unit length is therefore calculated using (10.3):

𝑄 𝑤𝑖𝑛𝑑,𝑚𝑎𝑥 = 𝑞𝑚𝑎𝑥 · 𝐶𝑥𝑐 · 𝐺𝑐 · 𝐺 𝐿 · 𝑑 · sin2 Ω = 680.42 · 1 · 1 · 0.0177 · sin2 90 = 12.04 𝑁 /𝑚 (C.4)

𝑄 𝑤𝑖𝑛𝑑,75% = 𝑞 75% · 𝐶𝑥𝑐 · 𝐺𝑐 · 𝐺 𝐿 · 𝑑 · sin2 Ω = 382.74 · 1 · 1 · 0.0177 · sin2 90 = 6.77 𝑁 /𝑚 (C.5)

Overhead line 9
 Appendix C. Mechanical

The ice load per unit length is obtained from (10.6):

−
𝑄𝑖𝑐𝑒 =9.82 · 10 3 · 𝛿 · 𝜋 · 𝑡 · (𝑑 + 𝑡 /1000) =
−3
9.82 · 10 · 900 · 𝜋 · 30 · (0.0177 + 30/1000) = 39.73 𝑁 (C.6)
/𝑚

C.2 Maximum horizontal tension

Once the loads per unit length have been obtained for the different meteorological conditions,
the horizontal tensions can be calculated so that their value does not surpass a threshold under
any of the hypotheses.

For this, the state change equation must be used in order to compare the tensions of the different
hypotheses. None of the conditions should reach a horizontal tension higher than 70% of the
conductor’s RTS:
𝐻𝑚𝑎𝑥 < 0.7 · 61340 = 42938 𝑁 (C.7)

The initial conditions for this calculation will be EDS conditions with a horizontal tension equal
to 22% of the conductor’s tensile strength:
◦
• 𝜃1 = 15 𝐶
• 𝑄 1 = 𝑄 𝑤𝑒𝑖𝑔ℎ𝑡 = 6.326 𝑁 /𝑚
• 𝐻1 = 0.22 · 61340 = 13494.8 𝑁

The first hypothesis to study will be the max wind one, with the following final conditions:
◦
• 𝜃2 = 10 𝐶
√
• 𝑄2 = 6.3262 + 12.042 = 13.6 𝑁 /𝑚
From equations (10.17), (10.18) and (10.19)

−4
7.454 · 1010 · 1.862 · 10 · (211.85 · 6.326)2
𝑎 = − 13494.8 24 · 13494.82 (C.8)
+
−5 −4
+ 7.454 · 1010 · 1.887 · 10 · 1.862 · 10 · (10 − 15) = −9100.8 𝑁

−4
𝑏 = 7.454 · 10 · 1.862 · 10 · (211.85 · 13.6)
10
= 4 8 1012 𝑁 (C.9)
3
. ·
24

𝐻 2 [𝐻2 + (−9100.8)] = 4.8 · 1012 (C.10)

2

Solving previous equation, the value of H2 obtained is 𝐻2 = 20510.7 𝑁 , which is lower than the
maximum allowed of 42938 N, meaning that the wind conditions are correct.

The second hypothesis will be the ice conditions hypothesis, with the final conditions:
◦
• 𝜃2 = −5 𝐶

Overhead line 9
 Appendix C. Mechanical

• 𝑄2 = 6.326 + 39.73 = 46.056 𝑁 /𝑚

Overhead line 9
 Appendix C. Mechanical

The values for a, b and 𝐻2 are:

−4
7.454 · 1010 · 1.862 · 10 · (211.85 · 6.326)2
𝑎 = − 13494.8 24 · 13494.82 (C.11)
+
−5 −4
+ 7.454 · 1010 · 1.887 · 10 · 1.862 · 10 · (−5 − 15) = −13029.38 𝑁

−4
7.454 ·21010 · 1.862 · 10 · (211.85 ·
46.056) 13 3
𝑏= = 5.5 · 10 (C.12)
𝑁
24

𝐻2 = 42905.86 𝑁 (C.13)

Since the value of 𝐻2 is lower than the 42938 N limit, the ice hypothesis is correct.

The third hypothesis to study is the combined loads one, with the following final conditions:
◦
• 𝜃2 = − 5 𝐶
√︁
• 𝑄2 (6.326 + 39.73)2 + 6.772 = 46.551 𝑁

The values for a, b and 𝐻2 are:

−4
7.454 · 1010 · 1.862 · 10 · (211.85 · 6.326)2
𝑎 = − 13494.8 24 · 13494.82 (C.14)
+
−5 −4
+ 7.454 · 1010 · 1.887 · 10 · 1.862 · 10 · (−5 − 15) = −13029.38 𝑁

−
𝑏 = 7.454 · 10 · 1.862 · 10 · (211.85 ·
10 4
13
𝑁3 (C.15)
46.551)2
= 5.62 · 10
24

𝐻2 = 43176.37 𝑁 (C.16)

In this case, the horizontal tension for the heavy load hypothesis is higher than the maximum
horizontal tension allowed. This means that the horizontal tension should be lowered so that,
under heavy load conditions, the horizontal tension is within the safety limits of the conductor.
Therefore, the initial conditions are changed to the following:
◦
• 𝜃1 = −5 𝐶
• 𝑄1 = 46.551 𝑁 /𝑚
• 𝐻1 = 42938 𝑁

These conditions will make sure that the heavy load hypothesis is respected while ensuring that
the previously studied hypotheses are also respected since the horizontal tension will be lower
with the new conditions (lowering the tension under one hypothesis will lower it for the rest of
them).

The maximum temperature hypothesis is not taken into account when trying to study if the
horizontal tension is respected, since this hypothesis will yield really low values of 𝐻2.
Overhead line 9
 Appendix C. Mechanical

C.3 Horizontal tension calculation

In the previous section, the most unfavorable initial conditions were established after studying
the different hypotheses, showing that the worst condition for this block is the one
corresponding to the heavy loads hypothesis.

Because the initial conditions were changed, the values for the horizontal tensions calculated
previous to the heavy loads hypothesis will have changed, meaning that they have to be recal-
culated with the new initial conditions.

In order not to repeat all the calculations, the final horizontal tensions with the new conditions
are obtained following the previous procedure with the heavy load’s conditions as the initial
conditions. The results are shown in Table C.2.

C.4 Catenary calculation

After obtaining the horizontal tensions, the catenary curve can be calculated for every
hypothesis (details in Section 10.3). The example calculation will be done for the heavy loads
hypothesis with the following values:

• 𝐻 = 42938𝑁
• 𝐴 = {0, 0, 0} , coordinates in meters
• 𝐵 = {245.4, 2.3, −4.4} , coordinates in meters
• 𝑚𝑐 = 46.056 𝑁 /𝑚 , considering the weight of the conductor and the ice

The span length can be calculated as:

√︃
𝑎= (𝐵.𝑥 − 𝐴.𝑥 )2 (𝐵.𝑦 − 𝐴.𝑦)2 = (245.4 − 0)2 (2.3 − 0)2 = 245.41 𝑚 (C.17)

√︃
The difference in altitudes is calculated from:
ℎ = 𝐵.𝑧 − 𝐴.𝑧 = −4.4 − 0 = −4.4 𝑚 (C.18)

The catenary constant c is calculated from Equation (10.9):

𝐻 = 42938
𝑐= = 932.3 𝑚 (C.19)
𝑤𝑐 46.056

The length of the curve is obtained from Equation (10.11):

√︂ √︄
. . 245.41 2
𝐿= 𝑎 2
= (−4.4)2 2 · 932.3 · sinh = 246.16 𝑚 (C.20)
ℎ2 + 2𝑐 · sinh 2·
+
2

The distance 𝑥𝐴 is then calculated according to (10.12):

𝐻
𝑥𝐴 =𝑐 · 𝑙𝑛 · (1 − exp (−𝑎/𝑐)) =
𝑤𝑐 · (𝐿 − (C.21)
ℎ)
−
932.3 · 𝑙𝑛

Overhead line 9
 Appendix C. Mechanical
· 1−
exp = −139.38 𝑚
42938 (−4.4)) 245.41
46.056 · (246.16 − 932.3

Overhead line 9
 Appendix C. Mechanical

𝑥𝐵 = 𝑥𝐴 + 𝑎 = −139.38 + 245.41 = 106.03 𝑚 (C.22)

The point where the maximum sag appears is calculated from Equation (10.14)

ℎ −4 . 4
𝑥 =𝑥 + 𝑐 · 𝑎𝑟𝑠ℎ = 0 + 932.3 · 𝑎𝑟𝑠ℎ = −16.61 𝑚 (C.23)
� �
� 245.

The max sag is therefore:

ℎ · (𝑥𝑎𝑐 − 𝑥𝐴) 𝑐𝑥𝐴 𝑥

𝑓 = + 𝑐 · cosh − cosh 𝑐 = (C.24)
−4.4 · (−16245.
.61 + 139.38) −139.38 = 8.089
cosh cosh 93
+ 93 −

All of the catenaries for the different hypotheses will be calculated for every span. By doing
this, the worst case scenarios for both, horizontal tension and clearances will be studied,
providing a good quality design.

The result for the different hypotheses are shown in Table C.2:

Table C.2: Sags and tensions

Temperature Loads Hor. tension Hor. tension

Hypothesis [ºC] [N/m] [N] [%] Max Sag [m]
Max Temp. 85 6.326 7469 12.17 6.38
EDS 15 6.326 13172 21.47 3.62
Wind Loads 10 13.600 20216 32.96 2.36
Ice Loads -5 46.056 42670 69.56 8.14
Heavy Loads -5 46.551 42938 70.00 8.09

C.5 Tower forces

The calculation of the forces in the towers will be performed considering one suspension tower
and one angle tower with one conductor per phase bundle and one circuit.

Table C.3: Inputs for tower forces calculation

Conductor
Wind Span Weight Span Weight Hor. tension
Tower [m] Hypothesis [m] [N]
[N/m]
Max wind
Heavy load 690.8 6.326 20216
Suspension 216.0 353.7 46.056 42938
Unbalance 352.9 46.056 42670
Cond. break 352.9 46.056 42670
Max wind
Heavy load 443.9 6.326 18391
Angle 216.8 279.66 46.056 42938
Unbalance 279.3 46.056 42645
Cond. break 279.3 46.056 42645

Overhead line 9
 Appendix C. Mechanical

Some additional data necessary for the calculation is presented:

• 𝑤𝑖𝑛𝑠 = 608.22 𝑁 , which is the weight of the insulator

• 𝑛𝑖𝑛𝑠 , which is the number of insulators per bundle, takes a value of 1 for the suspension
tower and 2 for the angle tower
◦
• 𝛼𝑎𝑛𝑔𝑙𝑒 = 3.27 , which is the deflection angle of the angle tower. This has to be divided by
two, since it takes into account the sum of the angle with both spans of the tower
• 𝑄 𝑤𝑖𝑛𝑑 = 12.043 𝑁 𝑚, which is the loads per unit length on the conductor of the
/
maximum wind hypothesis

C.5.1 Vertical forces

The vertical forces of one conductor bundle can be calculated as stated in Equation (10.20).
Two calculations will be performed as an example; firstly the vertical force in a suspension
tower with the wind hypothesis and secondly for the angle tower with heavy load hypothesis:

𝐹𝑉 = 𝑚𝑐 · 𝑔 · 𝑛𝑐𝑜𝑛𝑑 · 𝑎 𝑤𝑒𝑖𝑔ℎ𝑡 + 𝑛𝑖𝑛𝑠 · 𝑚𝑖𝑛𝑠 · 𝑔 = 6.326 · 1 · 690.8 + 1 · 608.22 = 4978.22 𝑁 (C.25)

𝐹𝑉 = 𝑚𝑐 · 𝑔 · 𝑛𝑐𝑜𝑛𝑑 · 𝑎 𝑤𝑒𝑖𝑔ℎ𝑡 + 𝑛𝑖𝑛𝑠 · 𝑚𝑖𝑛𝑠 · 𝑔 = 46.056 · 1 · 279.66 + 2 · 608.22 = 14097.02 𝑁 (C.26)

Where 𝑚𝑐 · 𝑔 is equal to the conductor weight, which considers ice if there is.

For the other conditions, the calculation will be the same, changing the values for the weight
span and the conductor loads.

C.5.2 Transversal forces

For the transversal forces, two components need to be calculated: the angle resultant and the
wind effect according to Equation (10.23). In this example, only the wind hypothesis will be
studied, for both suspension and tension towers.

For the suspension tower, the angle resultant will be disregarded, as no deflection angle can
exist for the designs provided by the software for this kind of towers. Therefore, only the wind
effect will be calculated using Equation (10.22):

𝐹 𝑤𝑖𝑛𝑑 = 𝑄 𝑤𝑖𝑛𝑑 · 𝑎 𝑤𝑖𝑛𝑑 · cos 𝛼 · 𝑛𝑐𝑜𝑛𝑑 = 12.043 · 216.0 · cos 0 · 1 = 2601.3 𝑁 (C.27)

The total transversal load on the suspension tower is calculated with Equation (10.23):

𝐹𝑇 = 𝐹𝛼 + 𝐹 𝑤𝑖𝑛𝑑 = 0 + 2601.3 = 2601.3 𝑁 (C.28)

For the angle tower, the angle resultant will have a value, calculated according to Equation (10.21):

𝐹𝛼 = (𝐻𝑇 1 + 𝐻𝑇 2) · sin 𝛼 = (20216 + 18391) · sin 3. = 1101.54 𝑁 (C.29)

2

Overhead line 9
 Appendix C. Mechanical

Since the angle tower has two neighbor spans that correspond to different blocks, the first span
and the second span will have different horizontal tensions. The first span has the same horizon-
tal tension as the suspension tower of study, whereas the second one has the value specified at
Table C.3.

The wind load is then calculated:

𝐹 𝑤𝑖𝑛𝑑 = 𝑄 𝑤𝑖𝑛𝑑 · 𝑎 𝑤𝑖𝑛𝑑 · cos 𝛼 · 𝑛𝑐𝑜𝑛𝑑 = 12.043 · 216.8 · cos 3. · 1 = 2609.86 𝑁 (C.30)
2

The total transversal force is:

𝐹𝑇 = 𝐹𝛼 + 𝐹 𝑤𝑖𝑛𝑑 = 1101.54 + 2609.86 = 3711.4 𝑁 (C.31)

C.5.3 Longitudinal forces

For the longitudinal forces calculation, two conditions will be considered: Unbalance hypothesis
for the suspension tower and Conductor break for the angle tower.

For the unbalanced condition with suspension towers, the corresponding percentage of horizon-
tal load that will appear due to the unbalance is equal to a 15%. Therefore, according to
Equation (10.24), the longitudinal force is calculated as:

𝐹𝐿 = 𝛿 · 𝐻 · 𝑛𝑐𝑜𝑛𝑑 · sin 𝛼 = 0.15 · 42670 · 1 · cos 0 = 6400.5 𝑁 (C.32)

For the angle tower, the conductor break hypothesis yields a horizontal force factor of 100%.
The longitudinal force is calculated as:

𝐹𝐿 = 𝛿 · 𝐻 · 𝑛𝑐𝑜𝑛𝑑 · sin 𝛼 = 1 · 42670 · 1 · cos 3. = 42652.6 𝑁 (C.33)

2

Overhead line 1