Optimizing Voltage for Power Transmission Efficiency

[skepticwulf]

Homework Statement

An electric power plant can produce electricity at a fixed power P, but the plant operator is free to choose the voltage V at which it is produced. This electricity is carried as an electric current I through a transmission line (resistance R) from the plant to the user, where it provides the user with electric power P' (a) Show that the reduction in power (P-P')due to transmission losses is given by (P-P'=P^2 x R / V^2 (b) In order to reduce power losses during transmission, should the operator choose V to be as large or as small as possible?

Homework Equations

P=I xV = V^2 / R = I^2 x R

The Attempt at a Solution

I think the power is lost due to long lines as R is increased by the length of the conductor. So due to formula(V^2 / R), we apply higher voltage to compensate this loss so that power is transmitted nevertheless as we wanted to be.So is there two Rs? R that leaves the plant, I mean right after the plant the length of the conductor is short so the R is small. And than comes R' which depends on the length of the conductor?
Do they apply small voltage to the power leaving the plant BUT then increase it before the transmission?
And I assume current is the same through the whole line.
How shall I calculate P-P' ?

 

[haruspex]

P and R are fixed. R goes all the way to the user, it's not in two parts. The user will never receive the full P since R>0.
If the plant produces power P at voltage V, what is the current? What current will the user get? What voltage will the user get?

 

[skepticwulf]

The current will be I=P/V and that will be constant , all the way to the user.
As P' is different due to R, and P'= I x V', which means V' will be different than V, smaller value that is.
Am I correct?

 

[gneill]

Do they apply small voltage to the power leaving the plant BUT then increase it before the transmission?

In general the voltage produced by the generators of a power plant is not necessarily the same as the voltage placed on the transmission lines. The voltage is adjusted using transformers. But for the purposes of this problem you should assume that any such transformers are considered to be part of the power plant -- all you can "see" coming from the plant is the transmission voltage and current.

And I assume current is the same through the whole line.
How shall I calculate P-P' ?

It might help if you draw a picture first, then label it with things you know and want to know. I'll give you a start:

 

[skepticwulf]

Got it, P-P'=I^2 x R, that's the lost power in lines.
As I=P/V
P-P'=P^2 xR / V

Thank you!