280 km, 60 Hz line: nominal-π vs long line

Given: $Z = 33 + j104\ \Omega$, $Y = j10^{-3}$ S (totals, so frequency is not needed), load 80 MW at 220 kV, 0.8 p.f. lagging. Per-phase values are used.

$$\begin{aligned} V_R &= \frac{220}{\sqrt3} = 127.02\angle 0^\circ\ \text{kV} \\ I_R &= \frac{80\times10^6}{\sqrt3\times220\times10^3\times0.8} = 262.4\angle -36.87^\circ = 209.95 - j157.46\ \text{A} \end{aligned}$$

(i) Nominal-Π method

$$\begin{aligned} ZY &= (33 + j104)(j10^{-3}) = -0.104 + j0.033 \\ A = D &= 1 + \frac{ZY}{2} = 0.948 + j0.0165 = 0.9481\angle 0.997^\circ \\ B &= Z = 109.1\angle 72.40^\circ\ \Omega \\ C &= Y\left(1 + \frac{ZY}{4}\right) = 9.74\times10^{-4}\angle 90.49^\circ\ \text{S} \end{aligned}$$ $$\begin{aligned} V_S &= AV_R + BI_R = (120{,}410 + j2096) + (23{,}304 + j16{,}638) \\ &= 143{,}720 + j18{,}734 = 144.93\angle 7.43^\circ\ \text{kV/phase} \\ V_{S(L)} &= \sqrt3 \times 144.93 = 251.03\ \text{kV} \\ I_S &= CV_R + DI_R = 200.58 - j22.09 = 201.8\angle -6.29^\circ\ \text{A} \end{aligned}$$

p.f. $= \cos(7.43^\circ + 6.29^\circ) = \cos 13.71^\circ = 0.9715$ lagging

$P_S = 3V_SI_S\cos\phi_S = 85.24$ MW (and $Q_S = 20.80$ Mvar)

(ii) Long-line (exact) method

$$\begin{aligned} \gamma l &= \sqrt{ZY} = 0.3303\angle 81.20^\circ = 0.05055 + j0.3264 \\ Z_c &= \sqrt{Z/Y} = 330.3\angle -8.80^\circ\ \Omega \\ A = D &= \cosh\gamma l = 0.9484 + j0.0162 = 0.9485\angle 0.98^\circ \\ B &= Z_c\sinh\gamma l = 107.2\angle 72.71^\circ\ \Omega \\ C &= \frac{\sinh\gamma l}{Z_c} = 9.828\times10^{-4}\angle 90.32^\circ\ \text{S} \end{aligned}$$ $$\begin{aligned} V_S &= AV_R + BI_R = 143{,}270 + j18{,}538 = 144.46\angle 7.37^\circ\ \text{kV/phase} \\ V_{S(L)} &= 250.23\ \text{kV} \\ I_S &= CV_R + DI_R = 200.98 - j21.11 = 202.1\angle -6.00^\circ\ \text{A} \end{aligned}$$

p.f. $= \cos(7.37^\circ + 6.00^\circ) = 0.9729$ lagging, $P_S = 85.21$ MW ($Q_S = 20.25$ Mvar)

Results and comment

• The two methods differ by only about 0.3 % in voltage, so for 280 km the nominal-Π model is reasonably accurate.
• The long-line method is exact. The error of the Π model grows with length, so the exact model should be used for lines well above 250 km.
• In both, $I_S < I_R$ because the line's charging current partly cancels the lagging load current.

Answer: Nominal-Π: $V_S = 251.03$ kV, $I_S = 201.8$ A, p.f. 0.9715 lag, $P_S = 85.24$ MW. Long line: $V_S = 250.23$ kV, $I_S = 202.1$ A, p.f. 0.9729 lag, $P_S = 85.21$ MW.