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January 30th, 2017, 06:06 AM   #1
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Increase in current as power is transmitted along a cable?

I have a situation where I have some (real) power demand at the end of a transmission line ($\displaystyle P_r$) for a given receiving-end voltage ($\displaystyle V_r$). For a line 150km in length the mathematics for calculating the sending-end voltage and current is laid out below:

Complex numbers...

Impedance:
$\displaystyle Z = R + 2\pi f L j$

where R is resistance of the cable, f is frequency of AC current and L is the inductance of the cable.

Admittance:
$\displaystyle Y = 2\pi f C j$

where C is the shunt capacitance of the cable.

Receiving power (including reactive component):
$\displaystyle S_r = P_r - P_r \tan \phi j$

where $\displaystyle \cos \phi$ is the receiving-end power factor (can be used to obtain $\displaystyle \tan \phi$).

Receiving current:
$\displaystyle I_r = S_r/V_r$

Define:
$\displaystyle A = 1 + \frac{YZ}{2}$
$\displaystyle C = Y\left(1 + \frac{YZ}{4}\right)$

Sending voltage and current:
$\displaystyle V_s = A V_r + Z I_r$
$\displaystyle I_s = C*V_r + A I_r$

Sending-power:
$\displaystyle S_s = V_s I_s^*$

$\displaystyle P_s = Re\{S_s\}$

For the following parameter choices...

$\displaystyle P_r$ = 1800 MW
$\displaystyle V_r$ = 400 kV
$\displaystyle \cos \phi = 0.9$, (so $\displaystyle \tan \phi = 0.4843$)
$\displaystyle R = 2.55 \Omega$
$\displaystyle L = 0.1289 $ H
$\displaystyle C = 5.05 \times 10^{-6} $ F

... I get the following results:

$\displaystyle I_r = 4500.00 - 2179.45 j$
$\displaystyle I_s = 4359.15 - 1475.70 j$
$\displaystyle |V_s| = 518$ kV

$\displaystyle P_s = 1863.75 $ MW


The value for $\displaystyle V_s$ is correct according to the textbook problem I am working through (Electric Power Systems 5th edition pp 120-121), so I think I've calculated everything correctly. However, the magnitudes of the current are

$\displaystyle |I_s| = 4602.16$ A
$\displaystyle |I_r| = 5000$ A

which just doesn't seem right to me... why is the sending current lower in magnitude than the receiving current? That is, why is the current increasing along the line? I would have expected the inductance and capacitance to cause a reduction in both current and voltage as the electrical energy is being transmitted along the cable.

Last edited by Benit13; January 30th, 2017 at 06:08 AM. Reason: typos...
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January 30th, 2017, 06:39 AM   #2
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January 30th, 2017, 06:52 AM   #3
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Haha!

According to wikipedia...
https://en.wikipedia.org/wiki/Electr...r_transmission

"Counterintuitive behaviors of medium-length transmission lines:

a) voltage rise at no load or small current

b) receiving end current magnitude can exceed sending end current"

So there it is... apparently the model can cause an increase in current. I'm not sure I'd refer to it as "counter-intuitive" personally, but hey ho!
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January 30th, 2017, 06:55 AM   #4
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Originally Posted by Benit13 View Post
Haha!

According to wikipedia...
https://en.wikipedia.org/wiki/Electr...r_transmission

"Counterintuitive behaviors of medium-length transmission lines:

a) voltage rise at no load or small current

b) receiving end current magnitude can exceed sending end current"

So there it is... apparently the model can cause an increase in current. I'm not sure I'd refer to it as "counter-intuitive" personally, but hey ho!
this shouldn't be very surprising. Your input and output networks can resonate with one another when their impedances are properly matched.
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January 30th, 2017, 07:08 AM   #5
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DC for the win!
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