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 understand why this is the correct dimension, let us analyze the variables and equations involved in the Newton-Raphson load flow study in polar form:

1. Bus Classification and Variables:
In a power system with N buses:

• There is 1 slack bus, where both the voltage magnitude and angle are specified. Thus, we do not need to solve for its voltage angle or magnitude.
• There are N_L load (PQ) buses, where both active power (P) and reactive power (Q) are specified. We need to solve for the voltage magnitude (V) and angle (δ) at these buses.
• The remaining buses are generator (PV) buses. The number of PV buses is given by N - 1 - N_L. At PV buses, the voltage magnitude is specified, so we only need to solve for the voltage angle (δ).

2. Dimensions of Mismatch Vectors and Unknown Vectors:
Based on the categorization above:

• Active Power Mismatch Vector (ΔP): Real power equations are written for all buses except the slack bus. Therefore, the dimension of ΔP is:
  $$(N-1)\times 1$$
• Reactive Power Mismatch Vector (��Q): Reactive power equations are written only for PQ (load) buses, as Q is not specified at PV and slack buses. Since there are N_L load buses, the dimension of ΔQ is:
  $$N_L\times 1$$
• Bus Angle Correction Vector (Δδ): The angles are updated for all buses except the slack bus. Therefore, the dimension of Δδ is:
  $$(N-1)\times 1$$
• Voltage Magnitude Correction Vector (ΔV): Voltage magnitudes are updated only for the PQ buses. Therefore, the dimension of ΔV is:
  $$N_L\times 1$$

3. Formulation of the Jacobian Matrix:
The linearized load flow equations relate these mismatch and correction vectors through the Jacobian matrix J:
$$\left[\begin{array}{c}\mathrm{\Delta P}\\ \mathrm{\Delta Q}\end{array}\right]=\left[\begin{array}{cc}H& S\\ M& R\end{array}\right]\left[\begin{array}{c}\mathrm{\Delta \delta }\\ \mathrm{\Delta V}\end{array}\right]$$

By expanding the matrix multiplication, the second row of equations gives:
$$\mathrm{\Delta Q}=M\mathrm{\Delta \delta }+R\mathrm{\Delta V}$$

For the matrix multiplication to be dimensionally consistent, the term M Δδ must yield a vector of the same dimension as ΔQ (which is N_L × 1):
$$\text{Dimension of}\mathrm{\Delta Q}=\left(\text{Dimension of}M\right)\times \left(\text{Dimension of}\mathrm{\Delta \delta }\right)$$
$$(N_L\times 1)=(\text{Rows of}M\times \text{Cols of}M)\times ([N-1]\times 1)$$

Equating the dimensions:

• The number of rows of M must be equal to the number of rows of ΔQ, which is N_L.
• The number of columns of M must be equal to the number of rows of Δδ, which is (N - 1).