Nominal-T relations, losses and Ferranti condition

In the nominal-T model, the total shunt admittance $Y = j\omega C$ is at the middle of the line and $Z/2 = (R + jX)/2$ is in each arm (per phase).

 IS     Z/2            Z/2     IR
 o-->--[R/2 jX/2]--+--[R/2 jX/2]-->--o
                   |
 VS              [ Y ] Ic   (VC)     VR
                   |
 o-----------------+-----------------o

Sending-end voltage and current

$$\begin{aligned} V_C &= V_R + \frac{Z}{2}I_R \\ I_S &= I_R + YV_C = YV_R + \left(1 + \frac{ZY}{2}\right)I_R \\ V_S &= V_C + \frac{Z}{2}I_S = \left(1 + \frac{ZY}{2}\right)V_R + Z\left(1 + \frac{ZY}{4}\right)I_R \end{aligned}$$

So $A = D = 1 + \frac{ZY}{2}$, $B = Z\left(1 + \frac{ZY}{4}\right)$, $C = Y$.

Active and reactive losses (3-phase)

Current $I_S$ flows in the sending half and $I_R$ in the receiving half:

$$\begin{aligned} P_{loss} &= 3\left(|I_S|^2 + |I_R|^2\right)\frac{R}{2} \\ Q_{loss} &= 3\left(|I_S|^2 + |I_R|^2\right)\frac{X}{2} - 3\omega C|V_C|^2 \end{aligned}$$

The last term is the reactive power generated by the line capacitance. Equivalently, $P_{loss} = P_S - P_R$ and $Q_{loss} = Q_S - Q_R$.

Condition for receiving voltage higher than sending voltage

At no load or light load ($I_R \approx 0$):

$$V_S = \left(1 + \frac{ZY}{2}\right)V_R \;\Rightarrow\; V_R = \frac{V_S}{1 + \frac{ZY}{2}}$$

With $Z = R + jX$ and $Y = j\omega C$:

$$1 + \frac{ZY}{2} = \left(1 - \frac{\omega CX}{2}\right) + j\frac{\omega CR}{2}$$

Neglecting the small $R$ term, $|V_R| > |V_S|$ when

$$\left|1 + \frac{ZY}{2}\right| < 1 \quad\Leftrightarrow\quad 0 < \frac{\omega CX}{2} < 2$$

which is always true for a practical line (and $R$ small). So at light load the receiving voltage rises above the sending voltage; this is the Ferranti effect. The rise is

$$|V_R| - |V_S| \approx \frac{\omega CX}{2}|V_S| = \frac{\omega^2LC\,l^2}{2}|V_S|$$

so it grows with the square of line length. With load, $V_R > V_S$ happens when the capacitive charging current outweighs the drop of the load current, i.e. when the load is below about SIL or is leading.