Kelvin's law and its modified form

Kelvin's law: the most economical conductor size is the one for which the annual cost of energy lost in the conductor equals the annual interest and depreciation on the part of the capital cost that varies with conductor size.

Proof

Let $a$ be the conductor cross-section.

1. The capital cost of the line is $P_1 + P_2a$. Here $P_1$ is the part that does not depend on size (towers, insulators, erection, approximately) and $P_2a$ is the conductor cost. The annual charge at a rate $r$ (interest + depreciation) is

$$C_1 = r(P_1 + P_2a) = k_0 + k_1a$$

2. The resistance is $R = \rho l/a$, so the annual energy loss is $3I^2R\,t$, where $t$ is the equivalent full-load hours, and its cost is

$$C_2 = \frac{k_2}{a}$$

3. The total annual cost is

$$C = k_0 + k_1a + \frac{k_2}{a}$$

4. For minimum cost:

$$\frac{dC}{da} = k_1 - \frac{k_2}{a^2} = 0 \;\Rightarrow\; k_1a = \frac{k_2}{a}$$

Also $\dfrac{d^2C}{da^2} = \dfrac{2k_2}{a^3} > 0$, so this is a minimum.

So the variable annual capital charge equals the annual cost of losses, which proves the law. The economical size is $a = \sqrt{k_2/k_1}$.

Why a modified form is used above 33 kV

Kelvin's simple law has several limits:

• It assumes the tower, insulator and erection costs do not depend on conductor size. At high voltage these costs dominate, and they do depend on conductor weight, diameter and sag.
• It ignores technical limits. Above 33 kV the minimum size is often set by corona (minimum diameter), by voltage regulation and stability, or by the thermal rating, not by economics.
• It assumes a constant load and constant interest and energy rates. Real loads vary, so the loss load factor must be used.
• It is hard to know the exact interest rate, depreciation and future energy price.

Hence for lines above 33 kV a modified Kelvin's law is used. The total annual cost of the whole line (towers + conductors + insulators + losses) is computed for several standard conductors that already meet the corona, thermal and regulation requirements. The conductor with the least total annual cost is then chosen.