In this article we will discuss about:- 1. Introduction to Transmission Lines 2. Conductor Configurations in Transmission Lines 3. Types of Conductors 4. Resistance 5. Skin Effect 6. Proximity Effect 7. Economical Size.
Contents:
- Introduction to Transmission Lines
- Conductor Configurations in Transmission Lines
- Types of Conductors Used in Transmission Lines
- Resistance of Transmission Line
- Skin Effect of Transmission Line
- Proximity Effect of Transmission Line
- Economical Size of Line Conductor-Kelvin’s Law
1. Introduction to Transmission Lines:
A transmission line consists of a set of conductors carrying electrical energy in bulk and transmitting it from power stations to primary substations. The conductors are parallel to each other and are carried on supports which provide insulation between the different conductors and between each conductor and earth.
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Figs. 3.1(a), (b) illustrate the transmission line in which line conductors are strung on insulators fixed to the cross-arms. Single circuit transmission line [Fig. 3.1(a)] carries 3 line conductors pertaining to three phases R, Y and B and one ground wire running over the top of the towers. Double circuit transmission line [Fig. 3.1(b)] carries 6 line conductors. These 6 line conductors constitute two separate circuits; each consists of three line conductors pertaining to phases R, Y and B. In case of double circuit transmission line two wires, known as ground wires, run over the top of the towers.
Transmission lines are basically electrical circuits having distributed constants— resistance, inductance, capacitance and shunt conductance. The shunt conductance is mostly due to leakage over the insulators and is so small that it can be neglected. These line constants, also called the line parameters, are uniformly distributed along the entire length of the line and are usually expressed as resistance, inductance and capacitance per unit length.
Concentration of all such parameters for the complete length of line at a single point is not possible. The performance of a transmission line depends to a considerable extent upon these parameters-series resistance causes a power loss in the conductor and so affects the transmission efficiency, series inductance mainly governs the power transmission capacity of the line, and shunt capacitance causes a charging current to flow in the line.
2. Conductor Configurations in Transmission Lines:
Several conductor configurations are possible, but three configurations are the most common i.e. horizontal configuration (or horizontal disposition of conductors), vertical configuration and triangular configuration.
There is no special advantage in using the symmetrical delta or triangular configuration [Fig. 3.3(a)] and in most cases flat horizontal or vertical configurations are employed from mechanical considerations, particularly when suspension insulators are used. In horizontal configuration, all the conductors are mounted over one cross-arm, as shown in Fig. 3.3(b).
Though such an arrangement of conductors needs supports of smaller height but needs a wider right of way. In certain congested areas where it is not possible to have horizontal arrangement of conductors, the conductors are placed in vertical formation (along the length of pole one below the other). The drawbacks of vertical formations are taller towers and more lightning hazards. There are places where both horizontal and vertical formations are applied.
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In unsymmetrical arrangement of conductors, the conductors are usually transposed at regular intervals in order to balance the electrical characteristics of various phases, and prevent inductive interference with neighbouring communication circuits.
Experience shows that a vertical configuration is the most economical for double circuit lines and horizontal or L-type configuration for single circuit lines.
3. Types of Conductors Used in Transmission Lines:
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The conductor is one of the important items as in a transmission and distribution system of electric power, the cost of the conductor material accounts for a major part of the total cost. So proper choice of conductor material and size of the conductor is of utmost importance.
The conductor material used for transmission and distribution of electric power must have the following characteristics:
(i) High electric conductivity i.e. low specific resistance.
(ii) High tensile strength in order to withstand the mechanical stresses.
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(iii) Low specific gravity in order to give low weight per unit volume.
(iv) Low cost in order to be used over long distances usually involved in transmission lines.
(v) Easy availability.
(vi) Should not be brittle.
No single conductor material meets all of the above requirements. Hence a compromise will have to be made between the cost and the desired electrical and mechanical properties in the selection of a conductor material for a given situation.
The most commonly used conductor materials for overhead lines are copper, aluminium, steel-cored aluminium, galvanized steel and cadmium copper.
All conductors used for overhead lines are preferably stranded in order to increase the flexibility. Solid wires, except of smaller sizes, are difficult to handle and when employed for long spans tend to crystallise at the points of support because of swinging in winds.
Stranded conductors usually have a central wire around which there are successive layers of 6, 18, 36 wires. For n layers, the total number of individual wire is 3n (n + 1) + 1. If the diameter of each strand is d then diameter of the stranded conductor will be (2n + 1) d. In the process of manufacture adjacent layers are spiralled in opposite directions so that the layers are bound together. The method of construction is called as ‘concrete lay’.
With conductors of large cross-section, however, another method known as ‘rope lay’ is sometimes employed in order to give more flexibility.
4. Resistance of Transmission Line:
Every electric conductor offers opposition to the flow of current and this opposition is called the resistance. The resistance of transmission line conductors is the most important cause of power loss (I2R) in a transmission line.
The ohmic resistance R of a conductor of length l and uniform x-section a is given by the expression:
where the resistivity or specific resistance of the conductor (ρ) depends not only on the conductor material but also on its temperature.
If ρ1 and ρ2 are the values of resistivity corresponding to two different temperatures t1 and t2 then,
ρ2 = ρ1 [1 + α (t2 – t1)] … (3.2)
where α is the temperature coefficient of resistance of the material. The value of temperature coefficient of resistance is also not constant but depends upon the initial temperature.
The temperature coefficient of resistance at any temperature t1 is given by the expression:
where α0 is the temperature coefficient of resistance at 0 °C.
In a single phase line or 2-wire d c line, the loop resistance is taken as double of resistance of either conductor but in 3-phase system the resistance per phase is the resistance of either conductor.
5. Skin Effect of Transmission Line:
The distribution of current over the cross-section of the conductor is uniform for dc only. An alternating current flowing through a conductor does not distribute uniformly but tends to concentrate near the surface of the conductor. In fact, in ac system no current flows through the core and the entire current is concentrated at the surface regions.
The result is that the effective area of the conductor is reduced causing an increase in its ac resistance. The effective ac resistance is usually referred as the effective resistance of the conductor. The phenomenon is called the skin effect as it causes concentration of current at the skin of the conductor.
The skin effect can be easily explained by considering a solid conductor to be composed of a large number of annular filaments, each carrying a fraction of total current. The flux linkages due to the filaments lying at the surface link the whole of the conductor while the flux set up due to the inner filaments does not link with the surface or outer filaments.
Thus the filaments near the centre are of larger inductance than that near the outer surface. The high reactance of the inner filaments causes the current to distribute in such a way that the current density is less in the interior of the conductor than at the surface. The distribution of current over the section will, therefore, be non-uniform, as shown in Fig. 3.5(b).
The skin effect depends upon:
(i) Type of material
(ii) Frequency
(iii) Diameter of conductor, and
(iv) Shape of conductor.
At low frequencies the effect is very small; in fact it is only of importance with high frequencies or with solid conductors of larger cross-section. For commercial frequency of 50 Hz or less the increase in effective resistance is inappreciable for solid copper conductors up to 1 cm in diameter; about 2.5 percent for 2 cm diameter and 8 per cent for 2.5 cm diameter.
In an aluminium wire the effect is the same as in a copper wire of equal conductivity. Thus since the resistivity of copper is 0.6 times that of aluminium, the increased resistance due to skin effect on an aluminium wire of a square mm in cross-section will be of the same percentage as on a 0.6 a square mm in copper wire.
The skin effect is much smaller with stranded conductors than with solid conductors. It increases with the increase of cross-section, permeability and supply frequency.
In practice stranded conductors are invariably used for transmission and distribution lines and hollow conductors for solid bus-bar. This is done in order to overcome the adverse effects of skin effect.
6. Proximity Effect of Transmission Line:
The inductance and, therefore, the current distribution in a conductor are also affected by the presence of other conductors in its vicinity. This effect is known as the proximity effect.
While considering the skin effect it was assumed that there is no other current carrying conductor nearby but when a current carrying conductor is nearby, its flux will link with the conductor under consideration and its effect to the nearer half of the conductor will be more than with the farther half.
If conductors carry currents in the opposite directions the magnetic fields set up will tend to cause an increase in the current density in the adjacent portions of the conductors while if the currents are in the same direction, the current density is increased in remote parts of the conductor.
So like skin effect, proximity effect affects the current distribution and results in an increase in the resistance of the conductor and decrease of self-reactance. Like skin effect, the proximity effect depends on the conductor size, frequency of the supply, resistivity and relative permeability of the material.
This phenomenon (i.e., proximity effect) is more pronounced for large conductors, high frequencies and close proximity. The magnitude of the effect, at normal supply frequencies, in the case of the wide spacing of conductors required for overhead transmission lines, is so small that it can be ignored. However, the proximity effect is pronounced in case of cables where the spacing between the conductors is small.
In the case of stranded conductors, each wire traverses alternately weaker and stronger portions of the magnetic field caused by the external current carrying conductor. Thus the average value of the field along the path of any wire remains the same, and assuming that the currents in the conductor follow to a large extent the paths of the individual wires, the proximity effect is substantially eliminated.
7. Economical Size of Line Conductor-Kelvin’s Law:
In case of a transmission system carrying bulk power over a long distance, the question of voltage regulation is unimportant; in some very long lines a regulation of 40 per cent is considered satisfactory. The only consideration is the economy. Hence the cross-section of the conductor of a transmission line (or feeder) is decided on the basis of its current carrying capacity and, where practicable, of minimum cost, both fixed and operating.
The fixed cost is on account of annual interest and depreciation on capital cost of conductors, supports and insulators and the cost of their erection in case of overhead systems. For underground systems the fixed cost is on account of annual interest and depreciation on cost of conductors, insulation and the cost of laying the cables.
For a particular voltage cost of insulation is practically constant and does not change with the x-section of the conductor, but the cost of conductors varies directly as the x-section of the conductor irrespective of the system of transmission. In case of overhead system the cost of supports and their crection partly varies as the x-section of the conductors and partly constant. Thus total annual fixed cost may be represented as Rs (P1 + P2 a) where P1 and P2 are constants and a is the area of x-section of conductor.
The annual running cost is on account of energy lost in the conductor due to its ohmic resistance i.e., I2 R losses; losses in insulating material and in metallic sheaths (for insulated cables). The losses in insulating material and in metallic sheaths are very small and in comparison to ohmic losses may be neglected for the low voltage, but at higher voltages these are considerable.
The ohmic resistance R is inversely proportional to area of x-section of conductor, and therefore for a given curve of demand for current throughout the year i.e. for a given annual load curve, the energy lost in the conductor will be proportional to the resistance and therefore, inversely proportional to area of x-section. Hence annual running cost on account of energy lost in transmission line may be represented as P3/a where P3 is a constant.
∴ Total annual cost, C = P1 + P2 a + P3/a … (3.4)
This total annual cost will be minimum if the differentiation of it with respect to a is zero
i.e. if dC/da = 0
i.e. if variable part of annual cost on account of interest and depreciation on the capital outlay is equal to the annual cost of electrical energy wasted in the conductors, the total annual cost will be minimum and the corresponding size of conductor will be most economical. This statement is modified version of Kelvin’s law. Though theoretically it holds good but in actual working condition number of difficulties are faced in its application.
Practical Limitations to Application of Kelvin’s Law:
Though theoretically Kelvin’s law is true but in actual working, an economical cross-section of the conductor determined by Kelvin’s law may not suit because of the following factors:
(i) The assumption that the total annual cost on account of interest and depreciation on the capital outlay is of the form P1 + P2 a is strictly speaking not true. For example, neither the cost of cable dielectric and sheath in underground cables vary according to this simple law, nor the cost of lying.
(ii) The size of the conductor determined may be of such a small x-section that it may cause too much voltage drop in the line.
(iii) The size of the conductor determined may be too weak or too strong from mechanical consideration. In case the size determined is too weak, it is advisable to go to the higher size irrespective to economy.
(iv) In case of cables the sizes of conductors determined by Kelvin’s law usually gives higher current density thereby giving excessive heating. The only remedy is to decide the size of conductors in case of cables on the basis of current carrying capacity.
(v) The diameter of the conductor may be so small as to cause high corona loss.
(vi) Interest and depreciation on the capital outlay depends upon commercial conditions and estimated life of conductors and also the probable scrap value at the end of estimated life, which are not known with a fair degree of accuracy.
(vii) It is difficult to estimate the energy loss in the line without actual load curves, which are definitely not available at the times of estimation.
(viii) It is also difficult to estimate the cost per unit of the energy wasted in the line. The cost per unit of the energy wasted is not the same as that of cost of generation per unit since their cost per unit depends upon load factors which are different for the line losses and the generation.
(ix) In the case of cables there are sheath losses and with high voltages dielectric losses also. Dielectric loss occurs continuously therefore load factor of dielectric loss is 100%. Hence the cost per unit of energy lost: as dielectric loss is less than cost per unit supplying the line loss.
It is impossible to derive an exact formula for the conductor x-section which will take into account all the above factors.
In case of overhead lines the question of temperature rise is of less importance and there are no losses other than I2 R loss so cross-sections derived from Kelvin’s law have more chances of being acceptable. In the case of underground systems it is advisable to examine the available cross-sections above and below the calculated section and to ensure that temperature rise will not be excessive.