# Kelvin's Law in Power System

### Kelvin's Law in Power System

Kelvins Law:
The most economical area of conductor is that for which the total annual cost of transmission line is minimum.
This is called as kelvins law after Lord Kelvin who first stated in 1881.

The transmission line cost forms major part in the annual charges of a power system.
The cost is due to

1. Depreciation
2. Repair and maintenance
3. Loss of energy in the line due to its resistance
4. The cost towards the production of the lost energy is considered
• If we decreases the area of the conductor in order to reduce the capital cost, the line losses increase.
• Similarly, if we increase the conductor cross-section to save the cost towards copper loss in the line, the weight of copper increases and hence the capital cost will be more.

Because of the above reasons, it is difficult to find the economical size of the conductor. But it becomes easy with the help of kelvin's law.

In this post we will understand about the kelvin's law and limitations of the kelvin's law.

Assume
A = Cross section of conductor

C = total initial cost towards conductor

C is directly proportional to A

C ∝ A

C = PA

where P is a constant.

Let r be the annual rate of interest and depreciation.
The annual fixed cost C1 = Cr = PAr
Since line losses are inversely proportional to the area of the conductor
The annual cost on lost energy,

C2 = Q/A where Q is a constant.

Total annual cost C = C1 + C2

= PAr + Q/A

For C to be minimum,

C/dA = 0

Pr - Q/A2 =0

Pr = Q/A2

Pr.A2 = Q

A2 = Q/Pr

A = √ (Q/Pr)

The equation shows that
"The economical cross-section of the conductor is that for which the annual charge on the conductor equals the annual charge for the loss of energy in the
conductor".
This is known as Kelvin's law.

Limitations of Kelvin's Law
This law has many problems and limits as we are selecting the cross-section from an economical point of view.We did not consider the electrical behaviour of the line.

1. It is not easy to estimate the energy loss in the line without actual load curves, which are not available at the time of estimation.
2. Kelvin's law did not consider many physical factors like voltage regulation, corona loss, temperature rise etc.
3. The assumption that annual cost on account of interest and depreciation on the capital outlay is not 100% true.
4. The conductor size determined by this law may not be always practicable one.
5. The rates of interest and depreciation may vary from time to time.
6. The diameter of the conductor may be so small as to cause high corona loss.
7. The conductor may be too weak to stamp from mechanical point of view.
8. Cost of insulation in cables is assumed to be independent of the cross-section of the conductor which is only an approx. assumption.

Remember that
Kelvin's law should not be applied to underground cables and high voltage overhead lines. Kelvin's law may be successfully used for overhead lines of voltage below 30KV.