Transient thermal response of underground power cables

QuickField simulation example

In this example we build the thermal model of the high-voltage cable buried in the ground. Initially the cable is energized, but there is no current. Then the current reaches 100% of the rated value. The task is to calculate the conductor temperature variation over time. Cable dimensions and material properties are taken from IEC 60853-2 example F2 [1].

Problem Type
Plane-parallel multiphysics problem of AC Magnetics and AC Conduction coupled to Heat transfer.

Geometry
Cable length is 1 km. Sheaths are grounded at one end.

Given
Voltage U = 400 kV (r.m.s.), AC frequency is 50 Hz.
Load cycle: 0 A for many days; 1580 A for 24 hours.

IEC 60853-2 [1] gives the thermal resistivity value (K*m/W) of materials. In QuickField we specify materials thermal conductivity, that is reciprocal to the thermal resistivity.

A cable conductor consists of many wires. The wires occupy only part of the conductor's cross-section. As you see, the cable conductor diameter is 57.5 mm, that gives the conductor cross-section area to be π*57.5²/4 = 2596.7 mm², while the cross-section of all copper wires is only 2000 mm². In the model, we do not draw individual wires in the conductor. We represent the conductor as a solid copper conductor with a diameter of 26mm, but we increase the copper resistivity in order to have the same resistance as in the real cable: ρ' = ρ * 2597mm²/2000 mm².

The resistivity of conductors depends on the temperature.
Copper electrical resistivity ρ=17.241 μOhm*mm, reciprocal of the temperature coefficient of resistance 234.5 K.
Lead alloy electrical resistivity ρ=214 μOhm*mm, reciprocal of the temperature coefficient of resistance 230 K.
In QuickField, you can specify the tabulated data for the electrical conductivity σ(T) temperature dependence:

In AC conduction problem we should specify the electric potential amplitude, which is U * √2. AC conduction problem setup also requires the electrical conductivity of materials. We can take into account dielectric losses by specifying the apparent value of the conductivity [2].
Insulation permittivity 2.5, loss tangent tan(δ) = 0.001.
Apparent value of the conductivity σ = 2πf·εε₀·tan(δ) = 2*3.142*50*2.5*8.854e-12*0.001 = 6.95 pS/m.

Task
Calculate the temperature response of the cable.

Solution
Initially the cable is kept energized with zero current for many days. Only dielectric losses are present in this period. AC conduction problem is simulated to calculate dielectric losses. Then a steady-state heat transfer problem HT0 is simulated to calculate the temperature T₀.

Then the load is connected and the current starts to flow in the conductor. Current generates Joule heat losses in conductors Q₀. We simulate AC magnetic problem AC0 to calculate the Joule heat loss value.
Then we can run the transient heat transfer analysis HT_T and calculate the temperature rise dynamics.

Conductor resistance depends on its temperature. While solving the transient thermal problem we might like to correct the conductor losses value. We can divide the transient heat transfer process HT_T into a sequence of transient problems HT1, HT2, HT3,... where each step simulates a short period of time.
After one intermediate problem is simulated, we run AC magnetic analysis to calculate losses at thetemperature reached to this moment. And then the next step transient problem uses the dielectric losses, conductor losses and this temperature distribution as initial..
This approach allows you not only to adjust the conductivity to the temperature at each moment of the transient process but also change the current in the conductor in case the load conditions change.

Results
Dielectric losses are 12.1 W/km. Dielectric losses heat the cable up to 28 °C.
At rated current the conductor losses are 36.5 W/m, sheath losses are 2.1 W/m.

References:
[1] IEC 60853-2, Calculation of the cyclic and emergency current rating of cables. Part 2: Cyclic rating of cables greater than 18/30 (36) kV and emergency ratings for cables of all voltages.
[2] Wikipedia, Dielectric loss.

• Video: Cable dielectric losses and temperature. Watch on YouTube
  
• Video: Cross bonded cable. Watch on YouTube
  
• Video: Transient thermal response of underground power cables. Watch on YouTube
  
• Download simulation files (files may be viewed using any QuickField Edition).