G-S first iteration with PV bus

Bus 1 is the slack bus ($1\angle 0^\circ$), bus 2 a PV bus ($P_2 = 3$ pu, $|V_2| = 1.05$ pu) and bus 3 a PQ bus ($P_3 = -4$, $Q_3 = -2$ pu). Initial values: $V_2^{(0)} = 1.05\angle 0^\circ$, $V_3^{(0)} = 1\angle 0^\circ$. Q-limits are neglected.

Y-bus

$$\begin{aligned} y_{12} &= 1/j0.3 = -j3.3333 \\ y_{13} &= 1/j0.2 = -j5.0000 \\ y_{23} &= 1/j0.2 = -j5.0000 \end{aligned}$$ $$Y_{bus} = \begin{bmatrix} -j8.3333 & j3.3333 & j5.0000 \\ j3.3333 & -j8.3333 & j5.0000 \\ j5.0000 & j5.0000 & -j10.0000 \end{bmatrix}\ \text{pu}$$

First Gauss-Seidel iteration

Iteration 1

$$\begin{aligned} \sum_{k=1}^{3} Y_{2k}V_k &= -j0.4167 \\ Q_2^{(1)} &= -\text{Im}\{V_2^{*}\textstyle\sum_k Y_{2k}V_k\} = 0.4375\ \text{pu} \end{aligned}$$ $$\begin{aligned} V_2^{(1)} &= \frac{1}{Y_{22}}\left[\frac{P_2 - jQ_2}{V_2^{*}} - Y_{21}V_1 - Y_{23}V_3\right] \\ \frac{P_2 - jQ_2}{V_2^{*}} &= \frac{3.0000 - j0.4375}{1.0500} = 2.8571 - j0.4167 \\ \textstyle\sum_{k\ne 2} Y_{2k}V_k &= j8.3333 \\ V_2^{(1)} &= \frac{(2.8571 - j0.4167) - (j8.3333)}{-j8.3333} = 1.0500 + j0.3429 \\ \delta_2 &= 18.0834^\circ \Rightarrow V_2^{(1)} = 1.0500\angle 18.0834^\circ = 0.9981 + j0.3259\ \text{(magnitude reset to }1.05\text{)} \end{aligned}$$ $$\begin{aligned} V_3^{(1)} &= \frac{1}{Y_{33}}\left[\frac{P_3 - jQ_3}{V_3^{*}} - Y_{31}V_1 - Y_{32}V_2\right] \\ \frac{P_3 - jQ_3}{V_3^{*}} &= \frac{-4.0000 + j2.0000}{1.0000} = -4.0000 + j2.0000 \\ \textstyle\sum_{k\ne 3} Y_{3k}V_k &= -1.6296 + j9.9907 \\ V_3^{(1)} &= \frac{(-4.0000 + j2.0000) - (-1.6296 + j9.9907)}{-j10.0000} = 0.7991 - j0.2370 \\ &= 0.8335\angle -16.5227^\circ\ \text{pu} \end{aligned}$$

Answer (end of first iteration): $Q_2 = 0.4375$ pu, $V_2 = 1.0500\angle 18.0834^\circ$ pu, $V_3 = 0.8335\angle -16.5227^\circ$ pu.