Long line by short model and vice versa

The long-line model (distributed parameters) is exact for any length. The short-line model keeps only the series impedance $Z$ and ignores the shunt capacitance ($A = D = 1$, $B = Z$, $C = 0$).

Long line modelled by the short-line model

A long line (above about 250 km) has a large shunt capacitance and large charging current, and the parameters are distributed. Using the short model:

• Charging current is ignored. The sending-end current is taken equal to the receiving-end current, which is wrong; at light load the real $I_S$ is mostly charging current.
• Sending-end voltage is wrong. With a lagging load the short model overestimates $V_S$ (no capacitive rise), so the voltage regulation comes out too high.
• Ferranti effect cannot be predicted. At no load the short model gives $V_R = V_S$, whereas a long line actually has $V_R > V_S$.
• Losses and efficiency are wrong, since the true current varies along the line.
• Power factor at the sending end is wrong; the reactive power generated by the line is missing.
• Reactive compensation, insulation and protection design based on these results would be unsafe.

Example: for a 300 km, 220 kV line delivering 50 MW at 0.8 lag (Z = 40 + j125 Ω, Y = j10⁻³ S), the short model gives $V_S \approx 251$ kV at p.f. 0.75 lag, whereas the Π model gives about 238 kV at p.f. 0.99 lead.

Short line modelled by the long-line model

• The long-line equations are exact for all lengths, so the results are correct.
• For a short line, $\gamma l$ is very small, so $\cosh\gamma l \approx 1$, $Z_c\sinh\gamma l \approx Z$ and $\sinh\gamma l / Z_c \approx 0$. The answers become practically the same as the short-line model.
• The only drawback is more calculation (complex hyperbolic functions) for no real gain in accuracy.

Summary