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

$\begin{bmatrix} \Delta P\\ \Delta Q \end{bmatrix}=J\begin{bmatrix} \Delta \delta \\ \Delta V \end{bmatrix}, \text{where}\: J=\begin{bmatrix} H & S\\ M &R \end{bmatrix}$

The dimension of the sub-matrix $M$ is 

  1. $N_{L}\times \left ( N-1 \right )$
  2. $\left ( N-1 \right )\times \left ( N-1-N_{L} \right )$
  3. $N_{L}\times \left ( N-1+N_{L} \right )$
  4. $\left ( N-1 \right )\times \left ( N-1+N_{L} \right )$
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