Kelvin's law and its limitations

Kelvin's law: the most economical conductor cross-section 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 is proportional to the cross-section.

Proof

Let $a$ be the conductor cross-sectional area.

1. Capital cost of the line = $P_1 + P_2 a$.
   • $P_1$ is the fixed part: towers, insulators, erection.
   • $P_2 a$ is the part proportional to area: conductor material and the extra tower strength.
2. With a rate $r$ of annual interest plus depreciation, the annual capital charge is

$$C_1 = r(P_1 + P_2 a) = P_1' + P_2' a$$

3. Energy loss: resistance $R = \rho l/a$. With an equivalent loss current $I$ over a year of $t$ hours, the energy lost is $3I^2\rho l t / a$. At a cost of $k$ per kWh:

$$C_2 = \frac{P_3}{a}$$

4. Total annual cost:

$$C = P_1' + P_2' a + \frac{P_3}{a}$$

5. For minimum cost:

$$\begin{aligned} \frac{dC}{da} &= P_2' - \frac{P_3}{a^2} = 0 \\ \Rightarrow P_2' a &= \frac{P_3}{a} \end{aligned}$$

Also $d^2C/da^2 = 2P_3/a^3 > 0$, so this is a minimum.

So at the most economical area, the variable annual capital charge equals the annual cost of energy loss, which proves the law. The economical area is

$$a = \sqrt{\frac{P_3}{P_2'}}$$

and the economical current density is $J = I/a$.

 cost
  |\                       / total
  | \ energy loss       /
  |  \  (P3/a)  ___/__
  |   \     _--   /
  |    `-.__     / capital (P2'a)
  |         ``--/--.____
  +-------------+---------> area a
            a_economical

Limitations

1. Interest and depreciation are hard to estimate exactly. Costs, interest rates and energy prices change over the life of the line.
2. The load varies, so the loss (equivalent) current is uncertain. Load growth is not considered.
3. It considers economy only. It ignores the technical limits of voltage regulation, corona, thermal ampacity, stability and mechanical strength. The economic size may be too small to meet these limits.
4. The conductor cost is not exactly proportional to area. Standard sizes are discrete, and the installation cost is not linear.
5. It does not include the cost of the extra generation capacity needed to supply the losses.
6. The capital cost of towers does not depend linearly on conductor area, especially when wind and ice loading govern the design.

For these reasons Kelvin's law is used only as a first guide. The final conductor is chosen by technical checks plus a present-worth comparison of standard conductors.