# Kelvin’s Law – Economic Choice of Conductor Size In the case of a transmission system carrying bulk power over a long distance, the question of voltage regulation is unimportant. Kelvin’s Law used to calculate the economical choice of conductor size in power system.

The only consideration is economy. Because the cost of conductor material is major component of total cost of transmission line. Therefore, it is necessary to choose the most economic size of the conductor.

The determination of economical size of the conductor is paramount importance. The cost of conductor material generally forms a considerable part of the total cost of the transmission line.

Most economical conductor size is that for which the annual cost of a transmission line is minimum.

## Annual charge on capital outlay

It is on account of annual interest and depreciation on the capital cost of a conductor, supports and insulators, and their erection cost in case of overhead transmission line.

For the underground system, it is on account of annual interest and depreciation on the cost of a conductor, insulation, and the cost of laying the cables.

For a particular voltage, cost of insulation is practically constant and does not change with X-section of the conductor. But the cost of the conductor varies “directly” as X-section of the conductor, irrespective of the system of transmission.

In the case of the overhead system, the cost of supports and their erection partly varies as the X-section of conductor and partly constant.

Thus, the total annual charge on an overhead transmission line can expressed as:

Total annual charge = P1 + P2a

Where P1 and P2 are constant and a is the area of X-section of the conductor.

## Annual cost of energy wasted in the conductor

It is account of energy lost in a conductor due to its ohmic resistance. i.e. I2R losses, losses in insulating material, and metallic sheaths (for insulated cables).

The resistance of the conductor is inversely proportional to its X-sectional area, the energy loss due to ohmic resistance may be represented as;

The annual cost of energy wasted = P3/a

Where P3 is constant.

Total annual cost, C = (P1 + P2a) + P3/a

The value of C will be minimum if differentiation with respect to a is zero.

“In other words, the variable part of an annual charge should be equal to the cost of annual losses due to energy wasted in the conductor for most economical working.”

The above figure shows a graphical representation of Kelvin’s law.

### Though theoretically Kelvin’s law is true, but in practice, there is a certain limitation because of the following factors

• The size of conductor material determined may be of a small area of cross-section causing high voltage drop or high voltage regulation in the line.
• The size of the conductor determined may be weak or strong from a mechanical point of view.
• For a smaller cross-sectional area, the current density in the conductor is high which may give rise to excessive heating. The only remedy is to increase the cross-sectional area of the conductor.
• The diameter of the conductor may be small causing high corona loss.
• The rate of interest and depreciation may vary from time to time and from place to place. Thus, the other data remaining the same, the conductor of economically designed lines will have different cross-sectional areas at different times and in different countries.
• It is difficult to estimate energy loss in line without actual load curves, which aren’t available at time of estimation.
• In the case of cables, the sizes of conductors determined by Kelvin’s law usually gives a higher current density, thereby giving excessive heating. The only remedy is to decide the size of conductors in case of cables based on current-carrying capacity.
• It is also difficult to estimate the cost per unit of the energy wasted in the line. The cost per unit of energy wasted is not same as that of cost of generation per unit. Since their cost per unit depends upon load factors which are different for line losses and generation.

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