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 NL. 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:  $\left[\begin{array}{c}\mathrm{\Delta }P\\ \mathrm{\Delta }Q\end{array}\right]=J\left[\begin{array}{c}\mathrm{\Delta }\delta \\ \mathrm{\Delta }V\end{array}\right]$ where$J=\left[\begin{array}{cc}H& S\\ M& R\end{array}\right]$

The dimension of the sub matrix M is

The correct option is N_L × (N - 1).

To determine the dimension of the sub-matrix M, we need to understand the variables and equations involved in the Newton-Raphson load flow study in polar form.

Let the total number of buses in the system be N.
These buses are classified as follows:
1. Slack Bus: There is always 1 slack bus. The voltage magnitude (V) and phase angle (δ) are specified, so they do not need to be solved.
2. PQ Buses (Load Buses): The number of PQ buses is given as N_L. At these buses, both active power (P) and reactive power (Q) are specified, while voltage magnitude (V) and phase angle (δ) are unknown.
3. PV Buses (Generator Buses): At PV buses, active power (P) and voltage magnitude (V) are specified, while phase angle (δ) and reactive power (Q) are unknown.
The number of PV buses is N - 1 - N_L.

Now, let's look at the mismatch vectors and their dimensions:
1. Active power mismatch vector (ΔP): Active power equations are written for all buses except the slack bus. Thus, the dimension of ΔP is:
(N - 1) × 1
2. Reactive power mismatch vector (ΔQ): Reactive power equations are written only for the PQ buses (load buses) because reactive power is not specified at PV buses or the slack bus. Thus, the dimension of ΔQ is:
N_L × 1

Next, let's analyze the state variable correction vectors:
1. Phase angle correction vector (Δδ): Phase angles are updated for all buses except the slack bus (whose angle is fixed). Thus, the dimension of Δδ is:
(N - 1) × 1
2. Voltage magnitude correction vector (ΔV): Voltage magnitudes are updated only for the PQ buses (since they are fixed at PV buses and the slack bus). Thus, the dimension of ΔV is:
N_L × 1

The load flow equations are represented in matrix form as:

$\left[\begin{array}{c}\mathrm{\Delta P}\\ \mathrm{\Delta Q}\end{array}\right]=\left[\begin{array}{cc}H& S\\ M& R\end{array}\right]\left[\begin{array}{c}\mathrm{\Delta \delta }\\ \mathrm{\Delta V}\end{array}\right]$

Expanding the matrix equation, we get the relation for ΔQ as:
ΔQ = M·Δδ + R·ΔV

To satisfy the matrix multiplication and dimensional consistency:
The dimension of the product (M · Δδ) must be equal to the dimension of ΔQ, which is N_L × 1.
Since Δδ has a dimension of (N - 1) × 1, the sub-matrix M must have:
- Rows equal to the number of rows of ΔQ (which is N_L)
- Columns equal to the number of rows of Δδ (which is N - 1)

Therefore, the dimension of the sub-matrix M is N_L × (N - 1).