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# Transmission line length changes

## Transmission line length changes

(OP)
How much does the physical length of a transmission line change with temperature and physical loading?

### RE: Transmission line length changes

S=w*L^2/8/T where:
S = mid-span sag (m)
w = conductor weight (N/m)
L = horizontal span length (m)
T = conductor tension (N)
The conductor tension T is the tension at the low point of the cable
Usually To=0.2-0.3 NBL [Nominal Break Load]
w=Wc[Kg/km]*9.81/1000
The difference in distance between the straight line between the supports and the distance along the parabola arc (the stretched conductor length) is called the slack. For a level span the slack is given by:
K=8*S^2/3/L
K = slack (m)
S = mid-span sag (m)
L = span length (m)
As the temperature increases, the unstretched conductor length will increase by an amount equal to: Δ L = α*dT* S where:
α = the coefficient of thermal expansion
dT = the temperature increase in deg C
S = the span length in metres
wind load on the conductor will increase the apparent weight of the conductor resulting in an in increase in tension.
Ice build up on the conductor will increase the apparent diameter and weight of the conductor
Aging: conductor sag over time may increase due to the effects of strand settling in and
metallurgical creep.
The increase in tension will increase the cable length due to elastic stretch by an amount given by: dL=(T-To)/E/A where:
To = the initial tension in newtons
T = the final tension
E = the coefficient of elasticity
A = the cross section of the conductor in metres.
See-for instance:
NETWORK LINES STANDARD GUIDELINES FOR OVERHEAD LINE DESIGN-ERGON ENERGY
https://www.ergon.com.au/__data/assets/pdf_file/00...

### RE: Transmission line length changes

As a less mathematically rigorous answer: The change in length for a typical span is on the order of centimeters, the change in sag is on the order of meters.

### RE: Transmission line length changes

Hey HamburgerHelper, don't know if it will help or not, but here's a quote of what I wrote in a recent post entitled Sag Protection, since I misunderstood the intended subject. I reproduce it here since I know I for sure don't read every single eng-tips post made, so you may not have seen it either...

My utility has a conductor sag detection facility used to derive realistic continuous and limited time ratings for specific 230 kV circuits that are often very heavily loaded, especially during the summer months [these circuits collectively comprise a "flowgate"]. Ultrasonic proximity detectors are used in conjunction with known line loadings and local ambient temperatures to calculate very accurate effective wind speeds which can then be plugged into the thermal monitoring program to enable said lines to be loaded to their maxima without crossing the fateful line into overload and premature conductor aging.

Anecdote, as all the formulae in the world can't correct a lack of situational awareness:

The Toronto Transit Commission once decided that better lighting was needed in its vehicle parking lots, and to that end subbed out a contract to have newer, taller, more efficient light standards installed.

One of these parking lots was located within a 230 kV right-of-way running roughly parallel to the 401 freeway; the contractor duly took their measurements from the conductors to the ground, determined there was no violation of clearances or limits of approach, and late in the fall of one year completed the installation to the satisfaction of the TTC, got paid, and the matter was closed...supposedly.

The following summer, on a hot, sticky day with lots of air conditioning load, there was a contingency involving a companion circuit, leading to a much heavier loading on the circuit in question, resulting in significant but nevertheless not unexpected or unacceptable conductor sag...with the predictable result including high-profile customer interruptions.

What was interesting was to watch the fur flying after the fact, with the different entities accusing each other of failing to exercise the appropriate due diligence, arguments over who was going to pay the additional cost of relocating the light standard in question, whether shortening it and using different luminaires with appropriately different light throws was a viable alternative, etc., etc.

### RE: Transmission line length changes

(OP)
Baconlife,

I am getting thrown off by your comment. If the conductor sags meters in a span, the conductor length has to be in the least near the length to go down and come back up. When I used this equations, Δ L = α*dT* S I was seeing length changes on the order of 0.2%. How do you get something that you can physically see sag when it is heavy loaded but its length only changes a small fraction of a percent. If a span is 300 M, I am maybe looking at 1 meter in length change. Does physical tension account for what I see more than its length change due to it being hotter?

### RE: Transmission line length changes

#### Quote (HamburgerHelper)

the conductor length has to be in the least near the length to go down and come back up
Not really. Consider 7Anoter4's equation for slack. For a 300m span with 3m of sag, the slack in the conductor is only 8*3^2/3/300 = 0.08m. That's not anywhere near down 3m + up 3m. A small change in length translates to a large change in sag.

Increasing temperature causes an increase in the conductor length, but that increase lowers the tension, which reduces the conductor length. The equilibrium point between the increase from temperature and the decrease from lower tension determines the change in slack and sag. The answer to the OP's question in short is "not much".

### RE: Transmission line length changes

Or, to put jghrist's response in different terms, the change is less than the uncertainty of the original measurement.

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