# Performance of Transmission Lines, Line Parameters and Corona

 Question 1
A $50 \mathrm{~Hz}, 275 \mathrm{kV}$ line of length $400 \mathrm{~km}$ has the following parameters:

Resistance, $R=0.035 \Omega / \mathrm{km}$;
Inductance, $\mathrm{L}=1 \mathrm{mH} / \mathrm{km}$;
Capacitance, $\mathrm{C}=0.01 \mu \mathrm{F} / \mathrm{km}$;

The line is represented by the nominal $-\pi$ model. With the magnitudes of the sending end and the receiving end voltages of the line (denoted by $V_{S}$ and $V_{R}$, respectively) maintained at 275 $\mathrm{kV}$, the phase angle difference $(\theta)$ between $V_{S}$ and $V_{R}$ required for maximum possible active power to be delivered to the receiving end, in degree is ____ (Round off to 2 decimal places).
 A 42.36 B 64.88 C 83.64 D 98.25
GATE EE 2023   Power Systems
Question 1 Explanation:
We have,
$\mathrm{P}_{\mathrm{R}}=\frac{\mathrm{V}_{\mathrm{S}} \mathrm{V}_{\mathrm{R}}}{\mathrm{B}} \cos (\beta-\delta)-\frac{\mathrm{AV}_{\mathrm{R}}^{2}}{\mathrm{~B}} \cos (\beta-\alpha)$

At max. power,
$\delta=\beta$

where, $\beta=$ angle of T-parameter of $B$.

$\pi$-Model :

$[T]=\begin{bmatrix} 1+\frac{yz}{2} &z \\ y\left ( 1+ \frac{yz}{4}\right )& 1+ \frac{yz}{2}\\ \end{bmatrix}=\begin{bmatrix} A & B\\ C&D \end{bmatrix}$
$\therefore \quad B=z$
$=(0.035 \times 400)+j(2 \pi \times 50 \times 10^{-3} \times 400) = 126.44\angle 83.643^{\circ}\Omega$
 Question 2
The bus admittance $\left(Y_{\text {bus }}\right)$ matrix of a 3-bus power system is given below.

$\begin{bmatrix} -j15 &j10 &j5 \\ j10& -j13.5 &j4 \\ j5& j4 & -j8 \end{bmatrix}$

Considering that there is no shunt inductor connected to any of the buses, which of the following can NOT be true?
 A Line charging capacitor of finite value is present in all three lines B Line charging capacitor of finite value is present in line 2-3 only C Line charging capacitor of finite value is present in line 2-3 only and shunt capacitor of finite value is present in bus 1 only D Line charging capacitor of finite value is present in line 2-3 only and shunt capacitor of finite value is present in bus 3 only
GATE EE 2023   Power Systems
Question 2 Explanation:
From $Y_{\text {Bus }}$ matrix
\begin{aligned} & y_{10}=-j 15+j 10+j 5=0 \\ & y_{20}=j 10-j 13.5+j 4=j 0.5 \\ & y_{30}=j 5+j 4-j 8=j 1 \end{aligned}

Power system network :

Hence, option (A) and (C) will not be correct.

 Question 3
The geometric mean radius of a conductor, having four equal strands with each strand of radius $'r'$, as shown in the figure below, is

 A $4r$ B $1.414r$ C $2r$ D $1.723r$
GATE EE 2022   Power Systems
Question 3 Explanation:
Redraw the configuration:

$\therefore \; GMR=(r' \times 2r\times 2r\times 2\sqrt{2}r)^{1/4}$
Where, $r'=0.7788r$
Hence, $GMR=1.723r$
 Question 4
Two buses, i and j, are connected with a transmission line of admittance Y, at the two ends of which there are ideal transformers with turns ratios as shown. Bus admittance matrix for the system is:
 A $\begin{bmatrix} -t_it_jY & t_j^2 Y\\ t_i^2 Y & -t_it_jY \end{bmatrix}$ B $\begin{bmatrix} t_it_jY & -t_j^2 Y\\ -t_i^2 Y & t_it_jY \end{bmatrix}$ C $\begin{bmatrix} t_i^2 Y & -t_it_jY\\ -t_it_jY & t_j^2 Y \end{bmatrix}$ D $\begin{bmatrix} t_it_jY & -(t_i-t_j)^2Y\\ -(t_i-t_j)^2Y & t_it_jY \end{bmatrix}$
GATE EE 2020   Power Systems
Question 4 Explanation:

\begin{aligned} I&=Y(t_{i}V_{i}-V_{j}t_{j}) \\ I_{i}&=t_{i}I \\ &=t_{i}^{2}YV_{i}-t_{i}t_{j}YV_{j} \\ I_{j}&=-t_{j}I \\ &=-I_{i}t_{j}YV_{i}+t_{i}^{2}YV_{j} \\ \begin{bmatrix} I_{i}\\ I_{j} \end{bmatrix}&=\begin{bmatrix} t_{i}^{2}Y &-t_{i}t_{j}Y \\ -t_{i}t_{j}Y &t_{j}^{2}Y \end{bmatrix}\begin{bmatrix} V_{i}\\ v_{j} \end{bmatrix} \end{aligned}
 Question 5
A lossless transmission line with 0.2 pu reactance per phase uniformly distributed along the length of the line, connecting a generator bus to a load bus, is protected up to 80% of its length by a distance relay placed at the generator bus. The generator terminal voltage is 1 pu. There is no generation at the load bus. The threshold pu current for operation of the distance relay for a solid three phase-to-ground fault on the transmission line is closest to:
 A 1 B 3.61 C 5 D 6.25
GATE EE 2020   Power Systems
Question 5 Explanation:
$I_{f}=\frac{1}{Z_{Th}}=\frac{1}{0.2}$
=5 pu for 100% of line
Relay is operated for 80%
$Z_{f}=0.8\, Z_{t}\Rightarrow 0.8\times 0.2=0.16\, p.u.$
For 80% of line,
$I_{f}=\frac{1}{0.16}=6.25\: p.u.$

There are 5 questions to complete.