Economic choice of conductor size - Kelvin's law

As economy is one of the most important factors while designing any transmission line, the cost of required conductor material is a considerable part. Thus, it becomes vital to select a proper size of the conductor. The most economic design of a transmission line is for which the total annual cost is minimum. Total annual cost can be divided into two parts, viz. annual charges on capital outlay and running charges. Annual charges on capital outlay include depreciation, interest on the capital cost, maintenance cost etc.. The cost of energy lost during the operation is counted in running charges. Regarding this, there are two important points that must be noted -

• if the cross-sectional area of the conductor is decreased, the total capital cost of the conductor decreases but the line losses increase (resistance increases with the decrease in the conductor size, hence, I²R loss increases)
• whereas, if the cross-sectional area of the conductor is increased, the line losses decrease but the total capital cost increases.

Therefore, it is important to find the most economical size of the conductor. Kelvin's law helps in finding this.
[Also read: Economic choice of transmission voltage]

Kelvin's law for finding economic size of a conductor

Let, area of cross-section of conductor = a
annual interest and depreciation on capital cost of the conductor = C₁
annual running charges = C₂

Now, annual interest and depreciation cost is directly proportional to the area of conductor.
i.e., C₁ = K₁a
And, annual running charges are inversely proportional to the area of conductor.
C₂ = K₂/a
Where, K₁ and K₂ are constants.

Now, Total annual cost = C = C₁ + C₂

                 C = K₁a + K₂/a
For C to be minimum, the differentiation of C w.r.t a must be zero. i.e. dC/da = 0.
Therefore,

"The Kelvin's law states that the most economical size of a conductor is that for which annual interest and depreciation on the capital cost of the conductor is equal to the annual cost of energy loss."
From the above derivation, the economical cross-sectional area of a conductor can be calculated as,
a = √(K₂/K₁)

Graphical illustration of Kelvin's law

As the annual cost of conductor is directly proportional to size of the conductor, it is shown by the straight line C₁ in the figure. Annual cost of energy loss is shown by the curve C₂. The total annual cost curve is obtained by adding the curve C₁ and C₂. The lowermost point on total annual cost curve gives the most economical size of the conductor which corresponds to the intersection point of curve C₁ and C₂. So, here, the most economical area of cross-section of the conductor is represented by ox and the corresponding minimum cost is represented by xy.

Limitations of Kelvin's law

Although Kelvin's law holds good theoretically, there is often considerable difficulty while applying it in practice. The limitations of this law are:

1. It is quite difficult to estimate the energy loss in the line without actual load curves which are not available at the time of estimation.
2. Interest and depreciation on the capital cost cannot be determined accurately.
3. The conductor size determined using this law may not always be practicable one because it may not have sufficient mechanical strength.
4. This law does not take into account several factors like safe current carrying capacity, corona loss etc.
5. The economical size of a conductor may cause the voltage drop beyond the acceptable limits.

Modified Kelvin's law

The actual Kelvin's law does not count the cost of supporting structures, erection, insulators etc.. It only accounts for the capital cost of conductor and corresponding interest and depreciation. Also, for underground cables, the cost of insulation and laying is not considered in the actual Kelvin's law. To account for these costs and to get practically fair results, the initial investment needs to be divided into two parts, viz (i) one part which is independent of conductor size and (ii) other part which is directly proportional to the conductor size. For an overhead line, insulator cost is almost constant and the cost of supporting structure and their erection is partly constant and partly proportional to the conductor size. So, according to the modified Kelvin's law, the annual charge on capital outlay is given as, C₁ = K₀ + K₁a. where, K₀ is an another constant. The differentiation of total cost C w.r.t. to the area of conductor (a) comes to be same as derived above under the heading Kelvin's law.
The modified statement of Kelvin's law suggests that the most economical conductor size is that for which the annual cost of energy loss is equal to the annual interest and depreciation for that part of capital cost which is proportional to the conductor size.
[Also read: Economics of power generation]