Power transfer and pf of motor on infinite bus

Reading of data: $V_t = 2300$ V and $E_f = 3450$ V are taken as line values (the rating is 2300 V line), with $E_f$ at $+120^\circ$ as printed. Work in per unit on 10 MVA, 2300 V.

$$\begin{aligned} V_t &= \frac{2300}{2300} = 1\angle0^\circ\ \text{pu},\quad E_f = \frac{3450}{2300}\angle120^\circ = 1.5\angle120^\circ = -0.75 + j1.299\ \text{pu} \\ Z_{base} &= \frac{2300^2}{10\times10^6} = 0.529\ \Omega,\quad X_s = 0.9 \times 0.529 = 0.4761\ \Omega \\ I_{base} &= \frac{10\times10^6}{\sqrt3 \times 2300} = 2510.2\ \text{A} \end{aligned}$$

Power transfer

$$P = \frac{V_tE_f}{X_s}\sin\delta = \frac{1 \times 1.5}{0.9}\sin120^\circ = 1.4434\ \text{pu} = 14.43\ \text{MW}$$

Current and power factor

Using the motor equation $\vec V_t = \vec E_f + jX_s\vec I_a$:

$$\begin{aligned} \vec I_a &= \frac{\vec V_t - \vec E_f}{jX_s} = \frac{1.75 - j1.299}{j0.9} = -1.4434 - j1.9444 \\ &= 2.4216\angle -126.59^\circ\ \text{pu} = 6079\ \text{A} \\ \vec S &= \vec V_t\vec I_a^{*} = -1.4434 + j1.9444\ \text{pu} \end{aligned}$$

The real part is negative: since $E_f$ leads $V_t$, the machine is not taking power; it is delivering 14.43 MW to the bus, i.e. acting as a generator. Taking the generated current $-\vec I_a = 2.4216\angle 53.41^\circ$ pu, it leads $V_t$ by $53.41^\circ$:

$$\text{pf} = \cos53.41^\circ = 0.596\ \text{leading}$$

        Ef (1.5 pu)
          \
           \ at 120 deg from Vt
            \
             O-----------> Vt (1.0 pu)
              \
               \  Ia (motor convention)
                v at -126.6 deg
 jXs.Ia = Vt - Ef joins tip of Ef to tip of Vt
 Ia(gen) = -Ia leads Vt by 53.4 deg

Note: with $\delta = 120^\circ$ the machine is beyond the steady-state limit ($90^\circ$) and the current (2.42 pu) is far above rating, so this point is not a stable working point. If the angle was meant as $-120^\circ$ (motor action, $E_f$ lagging), the same magnitudes result: the motor absorbs 14.43 MW at a pf of 0.596 lagging.

Answer: Power transferred $= 1.443$ pu $= 14.43$ MW; power factor $= 0.596$ (leading as printed, i.e. generating; lagging if $E_f$ is at $-120^\circ$).