UNTRANSPOSED ASYMMETRICAL OVERHEAD DISTRIBUTION LINES - REACTANCE 2

[wildalaska]

I'm trying to model a 12.47 kV distribution system (in the SKM PTW program) with untransposed irregular flat crossarms; maximum line length is about 10 miles.

Typical spacing on flat crossarms is phase A 0", phase B (middle) 26", and phase C 88". No transposing - i.e. phase B stays in the middle. For some sections they did periodically swap phase B from one side of the pole to the other, so then B would be 62" from A and 26" from C, and in the line sections where they did this the line becomes an untransposed (asymmetrical) REGULAR line (in which case the line may be treated as having phase B in the middle, thus simplifying the math somewhat, as A & C become symmetrical with respect to B).

Problem is, PTW, like almost ALL reference sources, assumes that asymmetrical lines are transposed, and even the SKM unbalanced module has no model for untransposed lines.

Review of basics: In an asymmetrical transmission line which has been TRANSPOSED, the three phases are periodically rotated into each position to average the effects of magnetic field cancellation (these unbalances show up as mutual reactances in the impedance matrices). Because of unequal magnetic field cancellation in an untransposed line, phase B has the lowest reactance, followed by whichever phase is closest to B (of course, in a regular untransposed line A & C will have the same reactance).

The standard assumption of a transposed line results in reactances composed of two components: Xa and Xd, where Xa is often called the reactance of the conductor out to 1' and only depends on the conductor's properties (GMR), and Xd is called the "spacing factor" and is the reactance from 1' out to the line separation. For transposed lines the line separation is the GMD = cube root of D1D2D3 (where the D are the line separations), and of course Xd is the same for all 3 phases.

So why not do as most engineers and just ignore the error? I'm tempted. After all, it's on the primary so its effect on the secondary is expected to be small. Well, in the past I would have. But we are now in the era of ARC-FLASH and sometimes either more (by causing extremely intense arcs) or less (by causing overcurrent devices to clear so slowly as to allow the arc to persist too long) AFC can increase the hazard to the point there is no PPE available. I've seen both cases occur ON THE SAME LOW VOLTAGE LINE: Stiff at the beginning, mushy at the end. So one can't always just make assumptions that cause more or less AFC; more accuracy than past practice required seems needed (the IEEE 85% assumption really doesn't help much). So my first question to answer: Is the overhead line asymmetry significant enough that it should not be ignored? I'm thinking if its effect is less than 5% I may ignore it.

Derive asymmetrical spacing factors for phase A case: I know the Ph.Ds go crazy analyzing these types of things, but I'm just a working engineer trying to come up with a model to correctly solve a problem. So using the great old Westinghouse T&D Ref Book, and starting from their derivation of Xd for transposed lines (Chap. 3, pp. 37-39), I derived the following pos & neg seq reactances:

ASSUMPTIONS:
1. Load impedance is balanced and >> line impedance, so that phase currents are approximately balanced and symmetrical.
2. Source voltage is balanced and symmetrical.
3. Line resistance = 0 (since inductance is geometric, we can add R back in later).
4. Ignore zero sequence because all transformers have line-to-line primaries, and arc-flash is only a concern at the low voltage level.
5. Other assumptions are from the Westinghouse text, pp. 34-39.

PHASE A POSITIVE SEQUENCE CASE (i.e. CW rotation):
From Eq 19, p. 37, and being careful to preserve the vector notation (which, for the transposed case, Westinghouse could omit), we have
(Eq1) Ea - E'a = Ia jXa - Ib jXd12 - Ic jXd13
where Xa = self-inductance out to 1' and Xd = spacing factor.
Then,
(Eq 2) ZA+ = jXA+ = (Ea - E'a)/Ia = jXa - (Ib Xd12)/Ia - (Ic Xd13)/Ia
Now, from assumption 1, and remembering that if the phase currents lag the phase voltages by an angle a, this angle a is subtracted out, we note that
- Ib/Ia = 1 @+60 and - Ic/Ia = 1 @-60, so Eq 2 reduces to
(Eq 3) XA+ = Xa + Xd12@+60 + Xd13@-60
Where XA+ is the positive sequence reactance of phase A. Note that where Xd = Xd12@+60 + Xd13@-60, non-zero IM{Xd} implies the existence of a negative or positive (phantom) resistance. Result looks unphysical because X is expected to be just a magnitude, but note that if Xd12 = Xd13 this reduces to the Westinghouse result for symmetrical or transposed lines. If correct, seems to imply the existence of a resistive element, even though the lines were assumed to have no resistance.

PHASE A NEGATIVE SEQUENCE CASE (CCW rotation):
For the negative sequence case, the angle of Vb- = 120 degrees, and the angle of Vc- = 240 degrees. From assumption 1, and once again remembering angle a, we have (note that here the Iabc are neg seq; for notational simplicity we just omitted the "-" from Ia-, etc.)

- Ib/Ia = 1 @ -60 and - Ic/Ia = 1 @ +60, so Eq 2 reduces to
(Eq 4) XA- = Xa + Xd12@-60 + Xd13@+60
Where XA- is the negative sequence reactance of phase A. Again, note that where Xd = Xd12@-60 + Xd13@+60, IM{Xd} implies the existence of a negative or positive (phantom) resistance.


QUESTIONS:
1. Has anyone calculated these things and determined if the asymmetry is significant with regard to arc-flash?
2. Above, I derived the reactances for the pos & neg sequence cases, but the PTW unbalanced module only appears to have an ABC domain Z matrix. Of course, as reactance is purely geometric, line resistance can be added back in at the end (see example below). Also Xa is available from reference books. The conversion from my results to the Z012 matrix, and then from the Z012 to Zabc matrix, looks difficult. Anyone have any slick ideas?
3. Any comments?

Actual line data for 397.5 ACSR, using the above derivation:
0.387133217613171 + j0.608804776800476 ohms/mile = Phase A pos seq Z
0.167670491192792 + j0.587556145614177 ohms/mile = Phase B pos seq Z
0.222196291194038 + j0.661533893295274 ohms/mile = Phase C pos seq Z
0.130866782386829 + j0.608804776800476 ohms/mile = Phase A neg seq Z
0.350329508807208 + j0.587556145614177 ohms/mile = Phase B neg seq Z
0.295803708805962 + j0.661533893295274 ohms/mile = Phase C neg seq Z

I’m out in the wilderness, so any wisdom will be appreciated!

Wild Alaska

 

[jghrist]

This might make a good theoretical paper if you had the time and inclination, but it is not practical to consider an exact line model. There might be a discernable difference in voltage drop for a load flow study, but I doubt that the error in a fault current study would be significant.

Consider what other assumptions are normally made such as gap distance that have a much more significant effect on arc incident energy. Do you know the exact impedance of cables in conduit? The three phases don't lie in an equilateral triangle. The effect of magnetic conduit is based on imperical measurements made decades ago and is certainly not exact. How much time do you spend determining actual cable lengths inside a plant? Do you consider all possible distribution line configurations? Do you consider all possible combinations of motors running in the plant?

 

[wildalaska]

Jghrist, That may very well turn out to be the case, but I think I should characterize it before ignoring. Regarding all the uncertainty in data, even though it's there, I have done several failure investigations of events on large industrial power systems and in every case that I can recall, the models predicted reality (to the extent we had comparative data). In the cases where the modeling was done with Dapper/Captor or PTW, I had done extensive "reality check" tweaking to the standard SKM models. So even though device manufacturer TCC data probably has enough uncertainty to drive Heisenberg crazy, the results still predicted reality. My philosophy on using computers is that’s why one uses them: To do the hard detailed calcs. I can usually do the rough calcs by hand.

WA

 

[healyx]

Hi wildalaska,

I have spent a bit of time investigating unbalanced/asymmetrical (untransposed) systems. I agree with you, everyone assumes transposition, but it is not followed through in practice. If you don't transpose, the results can be surprising.
The best source I found on this is a book by William Kersting (

http://www.amazon.com/Distribution-...1361230002&sr=8-3&keywords=william+kersting).

Kersting also has an old DOS program that does untransposed fault analysis. You can download it free. It is a bit clunky to use but probably provides the level of accuracy you are after.

 

[wildalaska]

Hey healyx, THANKS, I'll check it out. WA

 

[wildalaska]

Hey healyx,

The Kersting text looks like one I'd like to have, but in the meantime, is there a link for downloading his DOS program?

WA

 

[healyx]

Hi WA,

You can download RDAP form here:

http://www.zianet.com/whpower/whpc3.html

It is the educational version.

 

[dpc]

I agree with jghrist - you are assuming far more accuracy in the arc-flash calculation than is supported by the nature of the model. If you are calculating arc-flash on the overhead line itself, the requirements are covered rather crudely in the NESC and do not use IEEE 1584. If you are calculating arc-flash downstream of a step-down transformer, the impedance of the overhead line is a second-order effect and any consideration of transposition is even less important.

Due to all of the assumptions inherent in arc-flash calculations, I think getting within 10% of the actual incident energy would be the best that could be expected. Maybe within 20%.

But since you are stating line constants to about 12 decimal places, I suspect you won't be happy until you beat this horse to death, so have fun. Don't forget to account for the variation in fault current due to the variation in pre-fault conductor temperature.

 

[wildalaska]

dpc, your speculation about the OH line being "2nd order" on the secondary is probably correct and I commented to that effect in my original post. But I don't think assumptions like "it doesn't matter" belong in modern professional engineering practice. It should be characterized before being ignored. I wish I knew as much as you do. Because I don't, I have to seek knowledge. But one thing you may not know is that some spreadsheets insist on displaying a fixed number of digits for complex numbers - this may be evident by the blockish appearance of the digits. I didn't even think about the pre-fault temp. Thanks for the sugestion. WA

 

[healyx]

I jumped in on this thread because I have an interest in untransposed line analysis.
I agree with DPC that the effect of the upstream network on the LV side of a step-down transformer is negligible. It is also true that arc flash analysis is full of fudge factors anyway. That said, I do believe that if it isn't a big deal to add accuracy, then it can only improve the quality of the analysis.

Back to the question of transposition, I would like to make the point that that the absence of transposition in large LV cables can have a large affect on fault levels. Certainly not something I would regard as negligible. Yes, I said Low Voltage cables. No one bothers to transpose these and most LV engineers wouldn't know about transposition at all.

A 630mm2 (=1250 MCM) XLPE/PVC cable running 100m (328 ft) in a flat touching arrangement (untransposed) can see a fault level in one phase 20% higher than that predicted with the balanced case.

That said, you don't need to model that every time, just transpose the cables and use the simpler method. Problem solved.

 

[Janitor]

ATP/EMTP can do untransposed modeling and is free.

 

[Janitor]

ATP/EMTP is free and able to model untransposed lines and has several models for underground cables.

 

[wildalaska]

Hey Janitor, A 2nd unbalanced ap can't hurt and may help me to to clean up my act. Where can I download it? WA