# Performance of Transmission Lines, Line Parameters and Corona

 Question 1
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 1 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 2
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 2 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,
$I_{f}=\frac{1}{0.16}=6.25\: p.u.$
 Question 3
A three-phase 50 Hz, 400 kV transmission line is 300 km long. The line inductance is 1 mH/km per phase, and the capacitance is 0.01 $\mu F/km$ per phase. The line is under open circuit condition at the receiving end and energized with 400 kV at the sending end, the receiving end line voltage in kV (round off to two decimal places) will be ___________.
 A 418.85 B 256.25 C 458.45 D 369.28
GATE EE 2019   Power Systems
Question 3 Explanation:
\begin{aligned} V_s&= 400kV\\ l&=300km\\ L_1&=1mH/km/phase\\ C_1&=0.01 \mu F/km/phase\\ v&=\frac{1}{\sqrt{L_1C_1}}\\ &=\frac{1}{\sqrt{1 \times 10^{-3} \times 0.01 \times 10^{-6}}}\\ &=3.16 \times 10^5 km/sec\\ \beta '&=\frac{2 \pi f l}{v}\\ &=\frac{2 \pi \times 50 \times 300}{3.16 \times 10^5}=0.29\\ A&=1-\frac{\beta ^2}{2}\\ &=1-\frac{(0.29)^2}{2}=0.955\\ V_R&=\frac{V_S}{A}=\frac{400}{0.955}=418.85kV \end{aligned}
 Question 4
A three-phase load is connected to a three-phase balanced supply as shown in the figure. If $V_{an}=100\angle 0^{\circ} V$, $V_{bn}=100\angle -120^{\circ}V$ and $V_{cn}=100\angle -240^{\circ} V$ (angles are considered positive in the anti-clockwise direction), the value of R for zero current in the neutral wire is ___________$\Omega$ (up to 2 decimal places). A 5.77 B 2.45 C 4.75 D 6.25
GATE EE 2018   Power Systems
Question 4 Explanation:
From the given voltages,
\begin{aligned} I_R&=\frac{V_{RN}}{R}=\frac{100\angle 0^{\circ}}{R}\\ I_Y&=\frac{V_{YN}}{jX_L}=\frac{100\angle 120^{\circ}}{j10}\\ &= 10\angle -210^{\circ}\\ I_B&=\frac{V_{BN}}{-jX_C}=\frac{100\angle -240^{\circ}}{-j10}\\ &=10\angle -150^{\circ}\\ \text{For }\; I_N&=0\\ I_R+I_Y+I_B&=0\\ \frac{100}{R}+&10\angle -210^{\circ}+10\angle -150^{\circ}=0\\ R&=5.77\Omega \end{aligned}
 Question 5
For the balanced Y-Y connected 3-Phase circuit shown in the figure below, the line-line voltage is 208 V rms and the total power absorbed by the load is 432 W at a power factor of 0.6 leading. The approximate value of the impedance Z is
 A $33\angle -53.1^{\circ}\Omega$ B $60\angle 53.1^{\circ}\Omega$ C $60\angle -53.1^{\circ}\Omega$ D $180\angle -53.1^{\circ}\Omega$
GATE EE 2017-SET-2   Power Systems
Question 5 Explanation:
For star connection,
\begin{aligned} P_{3-\phi }&=\left | \frac{V_L^2}{z} \right | \cos \phi \\ Z&=\frac{V_L^2}{P_{3-\phi }} \cos \phi \\ &=\frac{208^2 }{430} \times 0.6 \\ \cos \phi &=0.6 \text{ lead}\\ \phi &=53.13^{\circ}\\ \therefore \; Z&=60\angle -53.13^{\circ}\Omega \end{aligned}
 Question 6
Consider an overhead transmission line with 3-phase, 50 Hz balanced system with conductors located at the vertices of an equilateral triangle of length $D_{ab}=D_{bc}=D_{ca}=1m$ as shown in figure below. The resistance of the conductors are neglected. The geometric mean radius (GMR) of each conductor is 0.01m. Neglecting the effect of ground, the magnitude of positive sequence reactance in $\Omega$/ km (rounded off to three decimal places) is ________ A 0.572 B 5.721 C 0.289 D 2.892
GATE EE 2017-SET-2   Power Systems
Question 6 Explanation: \begin{aligned} X_L&=2 \pi f L\\&=2 \pi f \times 2 \times 10^{-7} \ln \left ( \frac{D_m}{D_s} \right )\\ &=2 \pi \times 50 \times 2 \times 10^{-7} \ln \left ( \frac{1}{0.01} \right )\\ &=2.89 \times 10^{-4}\Omega /m\\ X_L&=0.289 \Omega /km \end{aligned}
 Question 7
The nominal- $\pi$ circuit of a transmission line is shown in the figure. Impedance $Z=100\angle 80^{\circ}\Omega$ and reactance X=3300$\Omega$. The magnitude of the characteristic impedance of the transmission line, in $\Omega$, is _______________. (Give the answer up to one decimal place.)
 A 406.2 B 201.5 C 635.8 D 52.4
GATE EE 2017-SET-2   Power Systems
Question 7 Explanation:
\begin{aligned} Z&=100\angle 80^{\circ}\Omega \\ X&=3300\Omega \\ \frac{y}{2}&=\frac{1}{X}=\frac{1}{3300} \end{aligned} \begin{aligned} y&=\frac{2}{3300}=6.06 \times 10^{-4}\\ Z_C&=\sqrt{\frac{Z}{y}}=\sqrt{\frac{100}{6.06 \times 10^{-4}}}\\ &=406.2\Omega \end{aligned}
 Question 8
A source is supplying a load through a 2-phase, 3-wire transmission system as shown in figure below. The instantaneous voltage and current in phase-a are $V_{an}=220sin(100\pi t)$V and $i_{a}=10sin (100\pi t)$A, respectively. Similarly for phase-b the instantaneous voltage and current are $V_{bn}=220cos (100\pi t)$V and $i_{b}=10cos( 100\pi t$A, respectively The total instantaneous power flowing form the source to the load is
 A 2200W B $2200 sin^{2}(100\pi t)W$ C 440W D $2200 sin(100\pi t) cos(100 \pi t) W$
GATE EE 2017-SET-1   Power Systems
Question 8 Explanation:
\begin{aligned} V_{an}&=220 \sin (100 \pi t)V\\ i_a&=10 \sin (100 \pi t)A\\ V_{bn}&=220 \cos (100 \pi t)V\\ i_b&=10 \cos (100 \pi t)A\\ p&=V_{an}i_a+V_{bn}i_b\\ &=2200 W \end{aligned}
 Question 9
At no load condition, a 3-phase, 50 Hz, lossless power transmission line has sending-end and receiving-end voltages of 400 kV and 420 kV respectively. Assuming the velocity of traveling wave to be the velocity of light, the length of the line, in km, is ____________.
 A 148.68 B 224.45 C 294.52 D 326.86
GATE EE 2016-SET-2   Power Systems
Question 9 Explanation:
\begin{aligned} V_s&=AV_R\\ 400 &=A420\\ A&=\frac{400}{420}=0.9524\\ A&=1+\frac{YZ}{2}\\ &=1+\frac{(r+j\omega L)()g+j\omega C)}{2}\\ &\text{For lossless line,}\\ r&=0,g=0\\ \text{then,}\\ A&=1-\frac{(\omega C)(\omega L)}{2}\\ \beta l&=\sqrt{\omega L\omega C}\\ A&=0.9524=1-\frac{\beta ^2l^2}{2}\\ \beta l&=0.3085\\ \beta &=\frac{0.3085}{l}\\ \frac{V}{f}&=\frac{2\pi}{\beta }\\ \frac{3 \times 10^5}{50}&=\frac{2 \pi}{\left ( \frac{0.3085}{l} \right )}\\ l&=294.52km \end{aligned} \begin{aligned} L_1&=2 \times 10^{-7} \ln \frac{D}{r}\\ &=2 \times 10^{-7} \ln \left ( \frac{100}{0.7788} \right )\\ &=0.97\mu H/m\\ L_2&=1.05 \times 0.97=1.0185 \mu H/m\\ L_2&=2 \times 10^{-7} \ln \left ( \frac{\sqrt{D^2 \times 100}}{0.7788} \right ) \end{aligned} \begin{aligned} \ln \left ( \frac{\sqrt{D^2 \times 100}}{0.7788} \right )&=\frac{1.0185 \times 10^{-6}}{2 \times 10^{-7}}\\ &=5.0925\\ e^{5.0925}&= \frac{\sqrt{D^2 \times 100}}{0.7788} \\ D&=142.7cm=1.427m \end{aligned}