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GATE 2021 EE – Question 19

Power Systems · Per-unit quantities, Bus admittance matrix, Gauss-Seidel and Newton-Raphson load flow methods · 1 mark · Multiple choice

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. $H$, $S$, $M$, and $R$ sub-matrices of the Jacobian matrix $J$ as below:

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

The dimension of the sub-matrix $M$ is

  1. $N_L\times(N-1)$
  2. $(N-1)\times(N-1-N_L)$
  3. $N_L\times(N-1+N_L)$
  4. $(N-1)\times(N-1+N_L)$

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Correct answer: (A) $N_L\times(N-1)$

Explanation

$\Delta P$ has one entry for every non-slack bus, $N-1$ in all, and $\Delta Q$ has one for every PQ bus, $N_L$. The unknowns $\Delta\delta$ are $N-1$ in number and $\Delta V$ are $N_L$. The sub-matrix $M=\frac{\partial Q}{\partial\delta}$ therefore has $N_L$ rows and $N-1$ columns.