Model validity and nominal-π ABCD with phasor

Why the long-line model fits all lines but the short-line model does not

• The long-line model uses distributed parameters and solves the line equations exactly: $A = \cosh\gamma l$, $B = Z_c\sinh\gamma l$, $C = \sinh\gamma l/Z_c$. No approximation is made, so it is valid for any length. For a short line $\gamma l$ is small, so $A \to 1$, $B \to Z$, $C \to 0$, which is exactly the short-line result.
• The short-line model neglects shunt capacitance ($C = 0$, $I_S = I_R$). For medium and long lines the charging current is large, so this model gives wrong $V_S$, $I_S$, power factor and regulation, and cannot show the Ferranti effect. So it is valid only for short lines (up to about 80 km).

ABCD parameters of the nominal-Π model

Total series impedance $Z$ in the middle; $Y/2$ at each end.

 IS       IL    Z = R + jX        IR
 o-->--+-----[ R ]--[ jX ]-----+-->--o
       |                       |
 VS  [Y/2] Ic1           Ic2 [Y/2]  VR
       |                       |
 o-----+-----------------------+-----o

$$\begin{aligned} I_L &= I_R + \frac{Y}{2}V_R \\ V_S &= V_R + ZI_L = \left(1 + \frac{ZY}{2}\right)V_R + ZI_R \\ I_S &= I_L + \frac{Y}{2}V_S = Y\left(1 + \frac{ZY}{4}\right)V_R + \left(1 + \frac{ZY}{2}\right)I_R \end{aligned}$$ $$A = D = 1 + \frac{ZY}{2}, \quad B = Z, \quad C = Y\left(1 + \frac{ZY}{4}\right)$$

Phasor diagram (lagging p.f. load, $V_R$ reference)

1. $I_R$ lags $V_R$ by $\phi_R$; $I_{C2} = j\frac{\omega C}{2}V_R$ leads $V_R$ by 90°.
2. $I_L = I_R + I_{C2}$.
3. $V_S = V_R + I_LR + jI_LX$ ($I_LR$ parallel to $I_L$, $I_LX$ 90° ahead of $I_L$).
4. $I_{C1}$ leads $V_S$ by 90°; $I_S = I_L + I_{C1}$.

       IC2 ^                    + tip of VS
           |         VS     .   |
           |            .       | IL.X
           |       .  d         |
   O---------------------------> VR
    \ \  phiR                   \ IL.R
     \  v IL                     +
      v IR       IS = IL + IC1 (IC1 is
                 90 deg ahead of VS)