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GATE 2017 EE – Question 20

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

A 3-bus power system is shown in the figure below, where the diagonal elements of $Y$-bus matrix are: $Y_{11} = -j12$ pu, $Y_{22} = -j15$ pu and $Y_{33} = -j7$ pu.

[Figure: Three buses. A generator is attached to Bus-1. A line of reactance $jq$ joins Bus-1 and Bus-2, a line of reactance $jp$ joins Bus-2 and Bus-3, and a line of reactance $jr$ joins Bus-1 and Bus-3. Bus-2 and Bus-3 have loads.]

The per unit values of the line reactances $p$, $q$ and $r$ shown in the figure are

Diagram for GATE 2017 EE question 20
  1. $p = -0.2$, $q = -0.1$, $r = -0.5$
  2. $p = 0.2$, $q = 0.1$, $r = 0.5$
  3. $p = -5$, $q = -10$, $r = -2$
  4. $p = 5$, $q = 10$, $r = 2$

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Show answer and explanation

Correct answer: (B) $p = 0.2$, $q = 0.1$, $r = 0.5$

Explanation

Each diagonal element is the sum of the admittances of the lines at that bus, and a line of reactance $x$ has admittance $-j/x$. So $\frac{1}{q} + \frac{1}{r} = 12$, $\frac{1}{q} + \frac{1}{p} = 15$ and $\frac{1}{p} + \frac{1}{r} = 7$. Adding gives $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 17$, so $\frac{1}{p} = 5$, $\frac{1}{r} = 2$ and $\frac{1}{q} = 10$. That gives $p = 0.2$, $q = 0.1$ and $r = 0.5$.