# Power Loss Calculation

#### jcbeck

##### Member
Really struggling to solve this question! How much power can be transmitted over a 30 mile long 33kV line which has a total reactance of 6.3 Ohms per phase and total resistance of 0.2 Ohms per phase? The answer is 57.62 MW per phase, but I can't get it to work out. Any assistance much appreciated! Thanks John

#### kingpb

##### Senior Member
Really struggling to solve this question! How much power can be transmitted over a 30 mile long 33kV line which has a total reactance of 6.3 Ohms per phase and total resistance of 0.2 Ohms per phase? The answer is 57.62 MW per phase, but I can't get it to work out. Any assistance much appreciated! Thanks John
Sounds like a homework problem!

To know how much power can be transmitted, you need to figure out the maximum current capability of the line, but to know that you have to know what the conductor size is. Then subtract your losses, and you should get your answer.

#### jcbeck

##### Member
Sounds like a homework problem!

To know how much power can be transmitted, you need to figure out the maximum current capability of the line, but to know that you have to know what the conductor size is. Then subtract your losses, and you should get your answer.
Yes, correct - it's part of my course. The problem is I haven't been given the conductor size! It's really left me scratching my head as to how to solve it!

#### kingpb

##### Senior Member
Those types of conductors are usually given r and x values in ohms/mi.

And I know the book I had for this type of problem had a conductor chart in the back...

with different conductors you can look up listing R, X, and C values. I wonder, if you converted the r and x to ohms per mile, and there was such a chart, what you might find?

#### Phil Corso

##### Senior Member
JCBeck...

3-phase or single-phase?

Regards, Phil Corso

3 phase

#### Smart \$

##### Esteemed Member
Yes, correct - it's part of my course. The problem is I haven't been given the conductor size! It's really left me scratching my head as to how to solve it!
Given the total reactance and total resistance per phase should be enough information to make conductor size nonessential...

...though a total resistance of 0.2 ohms per phase over 30 miles seems to indicate a very, very large conductor, even if paralleled multiple times over... :huh:

30 miles = 158.4kft
0.2ohms ? 158.4kft = 0.00126ohms/kft

That's less resistance than ten (10) paralleled 1,000kcmil copper conductors hmy:

Additionally, to determine effective Z, you also need to know the power factor.

There are definitely missing parameters to arrive at any determination... :happyyes:

#### Julius Right

##### Senior Member
You are speaking about Surge Impedance Loading[I guess] the power delivered by a lossless line to a load resistance equal to the surge (or characteristic) impedance Zc. You cannot deliver this power through your 30 miles transmission line due to a huge voltage drop.
This ideal power PSIL=VL-L^2/Zc. If we consider x=6.3 ohm as characteristic impedance of the line then Zc~6.3 ohm.PSIL=33^2/6.3=172.86 MW per all three phases or 172.86/3=57.62 MW per phase.

#### jcbeck

##### Member
Julius, thanks for that. My initial calculation was to include the resistance of 0.2 Ohms per phase with the reactance - why is the resistance ignored?

#### Besoeker

##### Senior Member
Julius, thanks for that. My initial calculation was to include the resistance of 0.2 Ohms per phase with the reactance - why is the resistance ignored?
I don't know whether or not it was ignored but it makes little difference to the calculation.

X 6.3
R 0.2
Z 6.303173804

Vl 33 kV
Vph 19.05255888
19052.55888 V
3022.692928 A
57590035.01 VA
57.59003501 MVA

Note that the answer is MVA, not MW.

#### Smart \$

##### Esteemed Member
I don't know whether or not it was ignored but it makes little difference to the calculation.

X 6.3
R 0.2
Z 6.303173804

Vl 33 kV
Vph 19.05255888
19052.55888 V
3022.692928 A
57590035.01 VA
57.59003501 MVA

Note that the answer is MVA, not MW.
How did you determine the effective Z???

#### Besoeker

##### Senior Member
How did you determine the effective Z???
Sqrt(X[SUP]2[/SUP]+R[SUP]2[/SUP])

#### Smart \$

##### Esteemed Member
Sqrt(X[SUP]2[/SUP]+R[SUP]2[/SUP])
I've seen this before when power factor is unknown. I don't understand. Can you explain, briefly?

#### JDBrown

##### Senior Member
I've seen this before when power factor is unknown. I don't understand. Can you explain, briefly?
I'll take a crack at it. Resistance and Reactance are vectors situated at 90 degrees from one another -- resistance on the real axis and reactance on the imaginary axis. Their vector sum is impedance.

You can solve for impedance using the Pythagorean Theorem (Z[SUP]2[/SUP] = R[SUP]2[/SUP] + X[SUP]2[/SUP]), which is what Besoeker did. Or you can use the Power Factor, which is really just the cosine of the angle between the impedance vector and the real axis.

Perhaps my mad MS Paint skills will help:

#### Besoeker

##### Senior Member
I've seen this before when power factor is unknown. I don't understand. Can you explain, briefly?
I think it's a very poorly framed question.
Working back from the result, the (unstated) assumption appears to be a short circuit at the end of the 30 miles so we are looking only at the impedance of the conductors.

#### Smart \$

##### Esteemed Member
I'll take a crack at it. Resistance and Reactance are vectors situated at 90 degrees from one another -- resistance on the real axis and reactance on the imaginary axis. Their vector sum is impedance.

You can solve for impedance using the Pythagorean Theorem (Z[SUP]2[/SUP] = R[SUP]2[/SUP] + X[SUP]2[/SUP]), which is what Besoeker did. Or you can use the Power Factor, which is really just the cosine of the angle between the impedance vector and the real axis.

Perhaps my mad MS Paint skills will help:
View attachment 9331
I understand it as presented. At some past point in time, I knew this. But what I don't understand is how this might relate to a realistic effective Z. I don't feel like perusing texts to discover the reasoning... so let us take for example the effective Z formula given in the notes to Chapter 9 Table 9:

Z[SUB]e[/SUB] = R ? PF + X[SUB]L[/SUB] sin[arccos(PF)]

According to my calculations, to arrive at an effective Z of 6.303173804 ohms using R=0.2 and X = 6.3 would require a power factor of approximately 0.03. That seems quite unrealistic to me...???

#### gar

##### Senior Member
131112-2340 EDT

The original post question lacks adequate definitions.

The title is "Power Loss Calculation".

The second sentence is
"How much power can be transmitted over a 30 mile long 33kV line which has a total reactance of 6.3 Ohms per phase and total resistance of 0.2 Ohms per phase?"

The first implies a measurement of power loss. Insufficient information.

The second implies maximum power transfer from the source to the load. This occurs when the source resistance equals the load resistance. Assuming the source reactance is inductive, then for maximum power transfer to the load, the load needs to be 0.2 ohms in series with a capacitive reactance equal to the source inductive reactance. This amounts to an extremely large amount of power. Clearly not the question for the answer given.

The original post needs to be rewritten with a correct statement of the question. It appears that the correct question may be something like --- what is the short circuit input volt-amperes from one line to neutral? This has nothing to do with maximum power transfer over a 30 mile transmission line.

Maximum power transfer will occur somewhere between a short circuit load and no load.

Without the capacitor in the load the maximum power transfer will occur around a load resistance of 6.3 ohms. I do not want to spend the time to do an exact calculation. Maximum power per leg is about 28.6 MW. Under these load conditions the load voltage is about 19*0.707 = 13.4 kV.

Check my math.

.

#### jcbeck

##### Member
Thanks to all - now have clear understanding and mental blockage now cleared!

#### Besoeker

##### Senior Member
According to my calculations, to arrive at an effective Z of 6.303173804 ohms using R=0.2 and X = 6.3 would require a power factor of approximately 0.03. That seems quite unrealistic to me...???
It might be plausible for an overhead transmission line where the conductors are spaces some distance from each other. You than have a single turn loop with a large area and an inductance of about 20mH seems feasible.

#### Smart \$

##### Esteemed Member
It might be plausible for an overhead transmission line where the conductors are spaces some distance from each other. You than have a single turn loop with a large area and an inductance of about 20mH seems feasible.
I didn't say impossible, just unrealistic. So just what large area and spacing are you implying? How does that relate to using sqrt(X?+R?) for Z[SUB]e[/SUB]?