Two generators have cost functions $F_1$ and $F_2$. Their incremental-cost characteristics are

$$ \frac{d F_1}{d P_1}=40+0.2 P_1 \text { and } \frac{d F_2}{d P_2}=32+0.4 P_2 $$

They need to deliver a combined load of 260 MW . Ignoring the network losses, for economic operation, the generations $P_1$ and $P_2$ (in MW) are

A 3-bus network is shown. Consider generators as ideal voltage sources. If rows 1,2 and 3 of the $Y_{h s}$ matrix correspond to bus 1,2 and 3 respectively, then $Y_{h s}$ of the network is

In the figure shown, self-impedances of the two transmission lines are $1.5 j \mathrm{pu}$ each and $Z_m=0.5 j \mathrm{pu}$ is the mutual impedance. Bus voltages shown in the figure are in pu. Given that $\delta>0$, the maximum steady state real power that can be transferred in pu from bus- 1 to bus- 2 is

Consider a power system consisting of $N$ number of buses. Buses in this power system are categorized into slack bus. $P V$ buses and $P Q$ buses for load flow study. The number of $P Q$ buses is $N_L$. The balanced Newton-Raphson method is used to carry out load flow study in polar form $H, S, M$ and $R$ are sub-matrices of the Jacobian matrix $J$ as shown below:

$$ \left[\begin{array}{l} \Delta P \\ \Delta Q \end{array}\right]=J\left[\begin{array}{l} \Delta \delta \\ \Delta \gamma \end{array}\right] \text {, where } J=\left[\begin{array}{ll} H & S \\ M & R \end{array}\right] $$

The dimension of the sub matrix $M$ is