Question & Answer

Given Data:

Sending end voltage (line-to-line), $V_s$ = 110 kV
Receiving end voltage (line-to-line), $V_r$ = 100 kV
Transmission line impedance per phase, $Z = 5 + j20$ ohms
Load power, $P = 50$ MW
Power factor (lagging), $pf = 0.9$

1. Calculate the line current flowing through the transmission line.

First, let's calculate the phase voltage and the current drawn by the load.

The line-to-line voltage at the receiving end $V_r$ is 100 kV, which translates to a phase voltage $V_{r\phi}$ :

V_{r\phi} = \frac{V_r}{\sqrt{3}} = \frac{100 \times 10^3}{\sqrt{3}} \text{ V}

The load power $P$ and power factor $pf$ are used to calculate the apparent power $S$ and the current $I_{r\phi}$ :

S = \frac{P}{pf} = \frac{50 \times 10^6}{0.9} \text{ VA}

I_{r\phi} = \frac{S}{\sqrt{3} \cdot V_{r}} = \frac{50 \times 10^6}{\sqrt{3} \times 100 \times 10^3 \times 0.9}

Now, let's compute these values.

The phase voltage at the receiving end is approximately 57,735.03 V (or 57.735 kV).

The apparent power $S$ is approximately 55.56 MVA.

The line current $I_r$ flowing through the transmission line is approximately 320.75 A.

2. Determine the voltage regulation of the transmission line.

Voltage regulation is calculated as:

\text{Voltage Regulation} = \frac{V_{s\phi} - V_{r\phi}}{V_{r\phi}} \times 100\%

To find $V_{s\phi}$ (sending end phase voltage), we use the voltage drop across the transmission line:

V_{s\phi} = V_{r\phi} + I_{r\phi} \cdot Z

Where:

$V_{s\phi}$ is the sending end phase voltage,
$I_{r\phi}$ is the current per phase.

Now, let's calculate the sending end voltage $V_{s\phi}$ and the voltage regulation.

The sending end phase voltage $V_{s\phi}$ is approximately 59,338.78 V with an imaginary component, indicating a phase angle (about 6,415.00j V), but the magnitude is the relevant value for voltage regulation.

The voltage regulation of the transmission line is approximately 3.38%.

3. Find the real power losses in the transmission line.

Real power losses in the transmission line $P_{loss}$ are given by:

P_{loss} = 3 \times I_{r\phi}^2 \times R

where $R = \text{Re}(Z)$ is the resistance per phase. Let's calculate the real power losses.

The real power losses in the transmission line are approximately 1.54 MW.

Summary of Results:

Line current flowing through the transmission line: 320.75 A.
Voltage regulation of the transmission line: 3.38%.
Real power losses in the transmission line: 1.54 MW.

Electrical engineering

Answer

Given Data:

1. Calculate the line current flowing through the transmission line.

2. Determine the voltage regulation of the transmission line.

3. Find the real power losses in the transmission line.

Summary of Results: