During a power failure, a domestic household uninterruptible power supply (UPS) supplies AC power to a limited number of lights and fans in various rooms. As per a NewtonRaphson load-flow formulation, the UPS would be represented as a

The figure shows the single line diagram of a 4-bus power network. Branches $b_1$, $b_2$, $b_3$, and $b_4$ have impedances $4z$, $z$, $2z$, and $4z$ per-unit (pu), respectively, where $z = r + jx$, with $r > 0$ and $x > 0$. The current drawn from each load bus (marked as arrows) is equal to $I$ pu, where $I \neq 0$. If the network is to operate with minimum loss, the branch that should be opened 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