Nominal-T derivation, phasor, losses, regulation

In the nominal-T model of a medium line, the total shunt admittance $Y = j\omega C$ is placed at the middle of the line, and half the series impedance ($Z/2$) is placed in each arm. Values are 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

Step 1: Voltage across the shunt (mid-point):

$$V_C = V_R + \frac{Z}{2}I_R$$

Step 2: Shunt current and sending current:

$$\begin{aligned} I_C &= YV_C = YV_R + \frac{ZY}{2}I_R \\ I_S &= I_R + I_C = YV_R + \left(1 + \frac{ZY}{2}\right)I_R \end{aligned}$$

Step 3: Sending-end voltage:

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

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

Phasor diagram (lagging load, $V_R$ reference)

1. $I_R$ lags $V_R$ by $\phi_R$.
2. $V_C = V_R + I_R\frac{R}{2} + jI_R\frac{X}{2}$.
3. $I_C$ leads $V_C$ by 90°; $I_S = I_R + I_C$.
4. $V_S = V_C + I_S\frac{R}{2} + jI_S\frac{X}{2}$.

                 IC ^          /| VS
                    |   VC   / IS.X/2
                    |   /|  /
                    |  / |IR.X/2
                    | /  /IR.R/2
   O-----------------------> VR
     \  \ phiR
      \   v IS  (IS = IR + IC)
       v
       IR

Power loss

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

Voltage regulation

No-load receiving voltage $= |V_S|/|A|$:

$$\%\text{VR} = \frac{|V_S|/|A| - |V_R|}{|V_R|}\times100, \quad |A| = \left|1 + \frac{ZY}{2}\right|$$

Efficiency

$$\eta = \frac{P_R}{P_R + P_{loss}}\times100 = \frac{3|V_R||I_R|\cos\phi_R}{3|V_R||I_R|\cos\phi_R + 3\left(|I_R|^2 + |I_S|^2\right)\frac{R}{2}}\times100$$