POWER PATH SUPPORT PORTAL

Calculations (formulas and equations) in (1) Sag-tension Report and (2) Support Report

1. Sag-tension Report

Catenary state equation

For medium spans: 

(1.1)

Where:

- a - span length,  a=const.

- y - angle between section direction and height difference:

, where:

- h - the vertical distance between the suspension points of the conductor

- a - span, 1 and 2 are the suspension points of the conductor

- a_{12 - a coupling that connects two suspension points}

Figure 1.1: Angle y representation (the angle between the coupling a₁₂ and horizontal a)

- g - specific conductor weight,

- s - tensile stress - depends on the material. So, if we have a conductor of 2 materials (e.g., Al/Ce), then this tensile stress is given as a function of the ratio of nominal cross sections of these two materials (e.g., for Al/Ce 240/60 the cross-sectional ratio of the materials is Al/Ce = 6). It occurs at low temperatures, while it decreases at high temperatures because the conductor elongates,

- g₀ - specific conductor weight, value depends on referent state,

- s₀ - tensile stress, value depends on referent state,

- t - current temperature,

- t_{0 - temperature at referent state,} 

- E – elastic modulus - is the tensile stress far from the tensile stress at which the material breaks. It is a constant that serves to establish the relationship between tensile stress and relative elongation in the zone of elastic deformation of the material. Relative elongation is the ratio of the change in length and the original length that occurs when a wire is stretched by some force.

Referent state is the state where conductor tensile stress has the maximum value. It can occur either at -20°C without additional load, or at -5°C with ice load. The tensile stress should have a maximum value in the temperature range (-20°C to +40°C), so that at other points the tensile stress is less than or equal to the maximum.

Initial conditions:

1) s_{0  = ksnd , t0} = -5°C, g₀ = g_R = g  + g_nd (it is adopted that it is s₀ < s_nd ), that is, k<= 1, so that the tensile stress at the higher point of suspension would not be higher ( s_F > s_nd, s_F - total tensile stress, which is obtained from formula: s_F = y   g , where y is a coordinate and g  is conductor weight). The coefficient k is determined by repeating the calculation several times until it is found, because with it s_F ≅ s_nd is achieved at the higher point of suspension. s_nd is the normally allowed tensile stress of conductor, i.e., the ground wire, and it must not be exceeded under normal conditions, i.e., at temp. of -20°C  without additional ice and at temp. of -5°C with additional ice. g_nd normal additional specific weight due to ice, which is estimated based on data collected in the previous period by the meteorological service as well as from experience gained on existing overhead lines whose routes are similar. If g_nd > g_ndmin is estimated, then it is calculated as g_nd = k   g_ndmin , where g_ndmin the minimum additional specific weight due to ice, which represents the mean value of the largest values that occur during the five-year period: 

, d - conductor diameter (mm) , s - the actual cross section of the conductor (mm²). The coefficient k is estimated from Table 1.1, while in practice values of  6,8 or more can be taken, depending on the area. Conductors should be calculated with g_R the resulting specific weight, because at temp. from -5°C ice occurs.

Table 1.1

2) s_{0  = ksnd , t0} = -20°C, g₀ = g . Depending on how we choose the initial conditions, we will calculate the equation of state (relation 1.1). If the first starting point is chosen (in most cases the maximum stress occurs at a temperature of -5°C with ice), then the system should be solved for t = -5°C  and g = g_R , until determined k , where the tensile stress at the higher point of suspension is s_F ≅ s_nd. After that, it should be solved for t = -20°C and g = g . If s_{F(-20°C )} ≤  s_nd, the starting point is well chosen, because no tensile stress can occur at any temperature in the range from -20°C to =+40°C  . If s_{F(-20°C )} ≥ s_nd , then the starting point is not well chosen, so another point should be chosen as the starting point. All this will be valid even if point number 2 is chosen as the starting point at the start. Then it should be solved by  t = -20°C and g = g , until the coefficient k is determined. With this, as in the previous case, ensures that at t = -20°C and without ice there is no s_F > s_nd . It is then resolved for t = -20°C  and g = g_R.  

Weather conditions

RULEBOOK (Official Gazette SFRJ)

Normal additional load due to ice is the largest additional load that occurs at a given place on average every 5 years and it is calculated according to the formula:

(1.2)

- d - diameter of conductor or ground wire (mm)

To estimate the additional load taken into account in the calculation, the ice coefficients obtained from the hydrometeorological service and the measured values on the existing overhead power lines and telecommunication lines along the projected route are taken into account:

1.0g

1.6g

2.5g

4.0g

This rule book defines ice load factor depending on the temperature with value:

 1, for the ambient temperature to -5°C ,

 1.6, for the ambient temperature from -5°C  to -20°C ,

 2.5 , for the ambient temperature from -20°C  to -30°C and

 4, for the ambient temperature below -30°C  .

Values other than those listed may be taken, but not less than 1.0g .

According to the Slovenian standard (EN 50341-2-21)

For normal additional load g_n , the highest additional load that occurs at a given place on average every 5 years is taken:

(1.3)

- d - diameter of conductor or ground wire (mm)

- f - ice load coefficient. Value may vary depending on the standards and regulations.

Ice load factor/coefficient is usually with values from 1 to 5. 

Additional note for ice load factor (read and check your standard). To estimate the additional load taken into account in the calculation, the ice coefficients obtained from the hydrometeorological service and the measured values on the existing overhead power lines and telecommunication lines along the projected route are taken into account.

Slovenian standards define ice load factor depending on the thickness of the ice sheet with a density of 900 kg/m3 where the ice load factor of:

 1, is for the ice thickness of 10 mm ,

 1.6, is for the ice thickness of 15 mm ,

 2.5, is for the ice thickness of  20 mm and

 5, is for the ice thickness of 30 mm .

The area with a coefficient of 1.6 is an area in which, based on weather conditions and many years of experience, only small loads of ice are created, which did not cause major damage to overhead lines.

The area with a coefficient of 1.6 is an area in which, based on weather conditions, geographical location and many years of experience, large loads of ice are created, which have already caused damage to overhead lines.

The area with a coefficient of 1.6 is an area in which, based on weather conditions, geographical location and many years of experience, very large loads of ice are created, which have already caused damage to overhead lines.

The density of ice in this standard is 900 kg/m3 , which is the amount of ice per cubic meter.

Critical span

Critical span is a span where conductor tensile stress at -20°C is the same as at -5°C with ice load.

(1.4)

Where:

- k = 1 - it has already been said how it is obtained (it is achieved with it at the higher point of hanging s_F ≅ s_nd)

- s_nd - nominal tensile stress,

- α - temperature coefficient,

- g_R - equivalent conductor specific weight (already mentioned):

- specific conductor weight with ice load,

- S – conductor cross section.

The critical span is calculated only from the equation for moderate (medium) spans, because the large ones are already bigger from the critical span. At large spans, the maximum tensile stress occurs at t = -5°C with ice, because the influence of additional specific weights due to ice is dominant. In contrast, at short spans, the impact of ice is smaller, and the maximum tensile stress can occur at t = -20°C .

After calculating the critical span, it is necessary to compare it with the span a. If a >a_kr, then the maximum tensile stress occurs at a temperature with ice, as already stated in the previous paragraph. In this case, the initial values are: s_{0  = ksnd, t0 = -5°C, g0 = gR = g  + gnd}. Otherwise, if a < a_kr,, the maximum tensile stress occurs at a temperature t = -20°C without ice. In this case, the initial values are: s_{0  = ksnd , t0} = -20°C, g₀ = g .

Critical span can be used to determine referent state.

For a > a_kr maximal tensile stress occurs at -5°C with ice load:

(1.5)

For a < a_kr  maximal tensile stress occurs at -20°C:

(1.6)

The previous relations enable the determination of the dependence s (t) by replacing and inserting the temperature.

If it is necessary to calculate a precise tensile stress at a temperature, then relations 1.5 and 1.6 are written in the form of polynomials: 

(1.7)

The coefficients of polynomials A and B are:

For a > a_kr

(1.8)

For a < a_kr 

(1.9)

B is not dependent on critical span.

(1.10)

Tensile stress equation can be solved iteratively.

Maximal conductor sag

Sag is the distance between the intersected points (points 1 and 2) of the vertical line (relative to the x axis) with the line of the conductor and the coupling (a₁₂), which connects the points of suspension of the conductor.

Maximal sag is defined at a point at catenary where tangent is parallel with attachment point. In Figure 1.2. is given the appearance of sag at an oblique span. The tangent is drawn through the point M and at that point the sag is maximal. Equation for sag at point M (x_M, y_M ):

(1.11)

For straight spans up to 1000m, this formula is used, which is accurate enough. For smaller spans, up to 300m, it is sufficient to use the first member of the relation 1.11, while the second member is taken only if it is a few centimeters. This second member is greater if the tensile stress is less, and the additional load is greater.

Figure 1.2: Maximal catenary sag at an oblique span

Relation 1.10 is not good enough for oblique spans, because the point M, where the sag is maximal, is not in the middle of the span. Then the abscissas of points 1 and 2 are not exact and this conditions the determination of the ordinate with an even greater error, which gives the wrong sag. Therefore, the sag of the oblique span can be calculated very accurately without determining the coordinates of points 1 or 2. This is done by turning the oblique span into a straight span by setting a new coordinate system (x_p, y_p) (Figure 1.3).

Figure 1.3: Oblique span in own  (x_, y) coordinate system and coordinate system (x_p, y_p) in which it is approximately straight

We see that the ordinate y_p passes through the point M where the sag is maximal. The span is not ideally straight in the coordinate system (x_p, y_p), because the point M is not in the middle of that span. But the sag f_p is quite close to the sag of the actual straight span in the coordinate system (x_p, y_p), because the curvature of the catenary is small in the point M area.

Table 1.2

Based on table 1.2 and based on the expression f_max = f_p / cos(y )  we have an expression that represents an approximate expression for the maximum sag of the oblique span:

(1.12)

Critical temperature

The maximum sag is the largest value of sag that occurs with a continuous change in temperature in the range from -20°C  to 40°C, i.e., the maximum sag may occur at temperature _tmax= +40°C or at t = -5°C  with ice. The term critical temperature is used to determine the maximum deflection.

At critical temperature catenary sag is equal to the sag at -5°C with ice load. Taking into account equation 1.10 and equality f_tKR = f_L (where: f_{tKR -} conductor sag at critical temperature and f_{L -} conductor sag with additional load due to ice) we have the following:   

If this is included in Equation 1.1, the expression for determining the critical temperature is obtained:

(1.13)

If the initial conditions s₀ = sL , _t0= -5°C , g₀ = g_R are included in this equation, the simplest expression for the critical temperature is obtained:

(1.14)

Where:

- t_kr - critical temperature,

- sL - tensile stress when ice appears.

For a > a_kr , relation 1.13 je directly applicable, because it is known in advance that s_{L  = ksnd} . For a < a_kr , tensile stress with ice load can be determined from the tensile stress equation. To solve, it is necessary to include the following conditions in relation 1.1: _t = -5°C, g = g_R, s = sL_, t = -20°C, g₀ = g and s_{0  = ksnd} .

If t_kr <40°C maximal sag occurs at _t = 40°C, otherwise, at _t = -5°C with ice load.

Ruling span

The tension field represents the distance between two adjacent tension poles. It strives for the spans to be equal, but this is not always possible. Therefore, if the tension field contains spans of different lengths, then when the temperature changes, the conductors in them will have different elongations. As a result, the suspension points of the conductors on the suspension poles will move and the insulators will take an oblique position. This all leads to the fact that relation 1.1 is extended by the article ∆L_a, which defines the elongation of the conductor due to the change of span (relation 1.1 for medium spans is derived for a = const.).

To determine tensile stress ruling span is used. A ruling span, also known as equivalent span or mean effective span (MES), is an assumed uniform design span which approximately portray the mechanical performance of a section of line between its dead-end supports. The ruling span is used in the design and construction of a line to provide a uniform span length which is a function of the various lengths of spans between dead ends. This uniform span length allows sags and clearance to be readily calculated for structure spotting and conductor stringing.

When the tensile stress is determined in the tension field, the sags on each span are calculated using expressions that apply to the individual range (relations 1.10 and 1.11).

In the catenary state equation span should be replaced by ruling span:

(1.15)

Where:

(1.16)

(1.17)

(1.18)

Relation 1.16. for straight spans directly gives the ideal a_i , while for oblique spans the product of the ideal span a_i and the cosine of the slope cos(yai) is obtained.

The sum of the projections of the absolute elongations of all spans of the tension field (relation 1.17) is equal to zero, because some spans increase and some shorten, while the suspension points on the first and last pole of the tension field are fixed.

Relation 1.18. represents how the cosine of the slope cos(yai) is obtained at the ideal span (cos(y) = a / a₁₂, at a span that is constant; see Figure 1.1 and the parameter y  explained in relation 1.1). 

To determine referent state use ruling span instead of actual span to compare to the critical span.

2. Support Report

Figure 2.1: Vertical and horizontal forces on support

V_x,V_y,V_z  are horizontal and vertical forces on support in X, Y, Z-axis direction, respectively. 

WIND FORCE - GENERAL EQUATION

Wind force Q_Wx due to horizontal wind blowing, at reference height, normal to any component is given by equation (5):

(2.1)

Where:

q_p(h) - maximum value of wind pressure,

h - referent height,

G_{x -} construction factor of the observed transmission line component,

C_{x -} force coefficient (aerodynamic coefficient), which depends on the shape of the observed component and

A_{x -} projection of the surface of the observed component on a plane normal to the wind direction.

Wind pressure is calculated as (Rule Book, Official Gazette SFRJ):

(2.2)

Where:

v - maximum wind velocity (m/s), which occurs on average every 5 years.

The wind pressure is applied to the base altitude zone from 0 to 40m and must not be less than 50daN/m². The obtained calculated values for p are increased to the first higher value from the following sequence:

60, 75, 90, 110, 130 50daN/m²

Increased values of wind pressure in the zone between 40m and 80m are taken according to Table 2.1:

Table 2.1.

Wind pressure according to Slovenian standard EN 50423-1(3) is defined as:

(2.3)

Where V is the wind velocity (m/s) that occurs on average every 5 years. The formula is valid for heights from 0 to 40m. The wind pressure is taken not to be below the value of 0,5 kN/m². The obtained calculated values are compared with the table below (Table 2.2) and the first higher value from it is used.

Table 2.2: Wind pressure depending on the zone and elevation

The new Slovenian standard EN 50341-2:2021 partially changed the procedure of wind pressure calculation, where, similar to the European standard, the basic wind velocity is first calculated (basic wind velocity is used according to wind zones), and based on it the wind pressure is obtained, whereby the terrain category is taken into account. For all transmission lines of voltage level above 45 kV, the II category of land is used, while for transmission lines of lower voltage levels, between II and III category is chosen.

Table 2.3: Value of wind pressure (N/m2) depending on the terrain category and wind zone

For transmission lines up to 15m high and voltage levels up to 45kV, the value of the mean wind pressure according to the wind zones is generalized (Table 2.3). If the heights of the poles are over 15m, the wind pressure is obtained by one of the following formulas:

- for the II category of the terrain:

(2.4)

(2.5)

- for the III category of the terrain:

(2.6)

(2.7)

Where:

q_{o -} wind reference pressure for a specific wind zone (zone 1 - 250 N/m², zone 2 – 390N/m² and zone 3 – 560N/m² ),

h - referent height.

The maximum value of wind pressure q_p(h) , at the reference height, which depends on the intensity of turbulence is given by equation (2.8):

(2.8)

Where:

I_V(h) - turbulence intensity,

q_h(h) - mean value of the wind pressure.

Turbulence intensity is defined at the reference height as the standard deviation of wind turbulence divided by the mean wind speed:

(2.9)

Where:

- reference heights above ground (h) depends on the transmission line component that is observed to be affected by the wind,

- terrain category (z₀) given in Table 2.4 and

- orographic factor (c₀).

Table 2.3: Value of wind pressure (N/m2) depending on the terrain category and wind zone

Wind force on support cross-arm

(2.10)

Where:

q – wind pressure,

G_c – structural factor, G_c = 1 – can be adopted or not, depending on the standard,

C_v – aerodynamics factor, C_v = 1 – can be adopted or not, depending on the standard,

d – conductor diameter,

L_{1 ,} L₂ – lengths of adjacent spans.

Structural factor is given according to expression (2.11):

(2.11)

Where:

k_p – the maximum factor is defined as the ratio of the maximum value of the variable part and the standard deviation,

Iv(h)  – turbulence intensity,

B² – environmental factor and

R² – resonance response factor, usually R² = 0.

The environmental factor is obtained as:

(2.12)

Where:

L_m – mean value of two adjacent spans given as: 

L(h)  – turbulent length scale (length of increased wind velocity) in m, on referent height h, which amounts to:

(2.13)

The structural factor, in general, decreases with increasing span length, and increases with increasing conductor reference height. Also, the structural factor decreases with increasing terrain category.

Factor G_c according to Slovenian standard EN 50423-1(3) for transmission lines up to 45kV. This is shown in table 2.5:

According to the Slovenian new standard EN 50341-2:2021 the structural factor G_c is calculated as follows:

(2.14)

Where:

k_rasp

• 1,0               for L_m ≤ 200m
• 0,6 + 80/L_m for L_m > 200m

k_kat

• 0,8               for the III category of the terrain
• 0,7               for the II category of the terrain 

k_cona

• 1,1               for I wind zone
• 0,9               for II wind zone
• 0,8               for III wind zone

Where:

- mean value of adjacent spans.

The aerodynamic coefficient C_c is determined in one of the following three ways:

• It is assumed that C_c = 1 , for used conductors,
• They were obtained from air tunnel tests,
• It is estimated based on the Reynolds number: 

(2.15)

Reynoldsov number is derived from the expression (2.16):

(2.16)

Where:

d  – conductor diameter,

v  – kinematic air viscosity ( v=15*10 - 6m²/s),

V(h)  – wind velocity.

For factor   according to Slovenian standard EN 50423-1(3) the value is used according to the table 2.6:

Wind force perpendicular to support cross-arm

(2.17)

Where:

α – alignment angle which was obtained as: α = θ₁ = - θ₂

Figure 2.2: Forces on support cross-arm

Where:

q_p(h)  – wind pressure,

h - referent height,

G_c  – structural factor of conductor,

C_c  – coefficient of force (aerodynamic coefficient) of the conductor,

d  – conductor diameter,

L_1, L₂  – lengths of adjacent spans,

φ – angle between wind direction and longitudinal axis of the console (Figure 2.2) and

θ_1, θ₂  - ( θ₁ + θ₂) * 1/2 = θ  is the route change angle.

Force on support cross-arm

(2.18)

Where:

s - conductor tensile stress,

S - conductor cross section and

α - alignment angle.

Force on support perpendicular to cross-arm

(2.19)

Where:

s - conductor tensile stress,

S - conductor cross section and

α - alignment angle.

Figure 2.3: Wind forces on conductor and consoles

As will be indicated and said later, F is the vertical force on the conductor and is obtained as: F = sS . These forces are obtained as a projection of the angle α, so that is why we have a sine or cosine.

Weight span

The weight span is used to calculate the vertical forces acting on the pole and consists of parts of two adjacent spans, which are defined by the distance from the pole to the vertices (bottom) of the conductor line (lowest point of the conductor) (Figure 2.4). Three cases of weight span:

Figure 2.4: Three cases of weight span: a) the vertices of both catenaries are within their spans; b) the vertices of one catenary are in the adjacent span; c) the vertices of both catenaries are in adjacent spans

A) Both lowest points are at their respective spans. The weight span for pole A is the distance between these vertices:

(2.20)

B) One lowest point is in its respective span (The vertices of both catenaries are within their spans). The weight span for pole B is:

(2.21)

Where a'₂ - extended span a₂ to the vertices of catenary.

This can be explained by the fact that the tensile stress at the suspension point on pole B is the same, regardless of where the lower suspension point on pole A is located. Pole A can theoretically be placed at any point of the extended span (e.g., points A' or A) and its weight span is:

(2.22)

The length a''₁ is subtracted from a'₁ , because the span on the right side of pole A loads the pole with the vertical component of the tensile force which acting upwards. In contrast, part a''₁ loads the pole B with a vertical force downwards. Depending on which part is larger, whether a'₁ or a''₁ a positive or negative value for a_gA is obtained, as can be seen from relation 2.22. If a''₁ > a'₁  the direction of vertical force at support (pole) A will be upwards, i.e., vertical force is negative. Then the insulator could be pulled out of the console (if it is supporting) or turned upwards (if it is hanging), so a counterweight should be placed on the insulator to compensate for the vertical component of the force acting upwards.

C) Both lowest points are in their adjacent spans. The weight span for pole B is certainly negative because both spans at the point of suspension on pole B generate vertical components of the force going upwards, and in this case, the counterweight should be placed.

Vertical force on support B has negative value.

Vertical force on support

The labels in the Figure 2.5 are: a - span, a_d - additional span, a_t - total span, h - the vertical distance between the suspension points of the conductor, 1 and 2 - the suspension points of the conductor, 4 - fictitious hanging point of total span, 3 - the vertices of the catenary, x,y - coordinate, s - horizontal tensile stress component, g  - specific weight of conductor, F_xx - the horizontal component of the tensile force of the conductor at the point (x,y) , F_yx - the vertical component of the tensile force of the conductor at the point (x,y), F_x - the total tensile force of the conductor at the point (x,y .