How to calculate power line loss with ohm's law and power equation?

Question

Hello I have a question about electricity. I know that transmitting power over long distance is better in AC. I also heard that the voltage needs to be high to reduce losses. I wanted to calculate this for myself.

Imagine there is a windmill that produces 100 kW of power. This 100 kW needs to be transported over a 1 km wire that has 1 ohm of resistance. We can transfer this 100 kW of power at 1kV or 10kV. But we are going to look which one is most beneficial.

When we send the 100 kW at 1 kV:

• This means we have 100 A (Power/Voltage) with a loss of 10 kW (Current²*Resistance).

When we send the 100 kW at 10 kV:

• This means we have 10 A (Power/voltage) with a loss of 0.1 kW (Current²*Resistance).

This proves that the losses are less when the voltage is high than when the voltage is low. However I noticed something odd. When we send the 100 kW at 1 kV I calculated that the current is 100 A via the power equation (100 000 / 1 000). But when I calculate the current via ohm's law I get 1 kA (1 000 * 1). What am I doing wrong?

EDIT: found it, can't use ohms law because not the full 1000 Volts is used on the transmission wire.

Answer 1

First note that DC is more efficient at transferring power long distances. AC is typically chosen because it is easier to change the voltage via a transformer.

An equation always needs a variable. If you arbitrarily specify all the values one value will never match the equation.

To back up for future readers Ohm's Law is V=I*R and for power P = V * I

In your scenario you arbitrarily specified all 4;
Power = 100kW
V= 1000V
I= 100A
R =1 ohm

Each time you use these equations you have to take a step back and understand the logical reason you are doing it. To further complicate things in the lower equations it seems that you calculated how much power would be consumed by the power line if it were a large heater. You have to consider the resistance the end users total power generation of the system.

Here is one of many possible use case scenarios for these equations: You have 1kV power source, a 1 ohm power line, and would like to transfer 75 amps of power to your users. How many watts of power will be lost in the power line? How much power to the users receive?

You first calculate your voltage drop in the power line with ohms law.

Voltage_Drop = Current * Line_Resistance
Voltage_Drop = 75amps * 1ohm
Voltage_Drop = 75volts
Power_Loss = 75volts * 75amps = 5625Watts

Voltage_At_Users = 1000volts - 75volts
Voltage_At_Users = 925volts
Power_Used = 925volts * 75amps
Power_Used = 69375Watts

We can confirm our math works by checking that power generated is the same as the power used plus the power lost.

Power_Generated = Power_Used + Power_Lost
Power_Generated = 69375Watts + 5625Watts
Power_Generated = 75000Watts

Power_Generated = 1000volts * 75amps
Power_Generated = 75000Watts

From here you can modify the line voltage and current and see how that changes your delivered power and losses.