Numerical simulation of overhead power line cooling in natural convection regime

Jožef Stefan Institute (JSI)Gregor KosecJure Slak

The Tenth International Conference on Engineering Computational Technology, 2018, Sitges, Barcelona, Spain

Scope

Cooling of conductor due to the natural convection

The most important cooling mechanism of a conductor, i.e.convection, is, in all standard Dynamic Thermal Rating (DTR)models, taken into account in terms of empirical relation thatare mostly based on data collected by Morgan in 1975.

( )2 W,

mJ I RQ T

=

W,

mS s TI DQ

=

4 4 W( )

mR B s s aQ D T T

= − −

ò

W( )

mC s aQ Dh T T

= − −

,a

hDNu

=

90Nu Re ,nB= Re ,f

uD

v=

2

20 20 20( ) (1 ( 20 C) ( 20 C) ),R T R T T = + − + −

( )( )a a a a a

p p

Tc c T T

t

+ =

v

0 =v

ref1 ( )ref T T T = − −b g

( )( )a a aPt

+ = − + +

vvv v bPhysical model

The domain of the problem is a cross-section of a power linethat is further separated into a steel core and aluminumconductor, and surrounding air

Numerical solutionThe solution procedure is divided in two time loops, namely theair and the power line loop. The involved Partial DifferentialEquations are discretized with RBF-FD method.

ResultsThe results of the simulation are presented in terms oftemperature and velocity magnitude contour plots, convergenceanalyses, and comparison of convective heat losses of simulatedresults to IEC, IEEE and CIGRE standards.

Numerical model vs. CIGRE/IEEE

Numerical solution of temperature and velocity fields around conductor

• Steady state achieved in order of 1 s

• Temporal discretization in order of 1 ms

CIGRE and IEEE assume empirical relation between temperature and cooling rate

In numerical model we solve thermo-fluid problem and compute cooling rate without any parameters.

2st

st st st st

p

Tc T

t

=

2al

al al al al

p J

Tc T q

t

= +

22

al aal a

s r

r r

T Tq q

− = +

n n

( ) ( )2 2

al aT r T r=

1 1

al stal st

r r

T T

=

n n

( ) ( )1 1

al stT r T r=

2

W,

m

s TS

Iq

=

( )( )4 4

2 2

W

malR B s aq T r T

= −

ò

( )2

3

W,

mal

al

j

Rq

S

I T =

Computation of heat generation and transport within the power line

• Steady state achieved in order of 10 min

• Temporal discretization in order of 10 s

CIGRE in IEEE assume homogenous conductor and assess the skin and core temperature from closed form solution of simplified problem.

In numerical model we compute 2D simulation of heat transfer and generation within the power line.

( )( )a

a a a a a a a

p p

Tc c T T

t

+ =

v

ref1 ( )a

ref T T T = − − b g

0 =v

( )( )a a aPt

+ = − + +

vvv v b

RadiationBoundary condition

Natural convection: Thermo-fluid problem

Steel part – heat transport

Aluminium part – heat transport and heat generation

Solution procedure

( )1

1( )i a

a a

a

t

= + − +

bv v v vv

( )2 1 1

1( )a

a a a

p

T T t T Tc

= + −

v

2 c i

air

pt

=

v

( )( ) ˆaP

n

= +

v b n

0pd

=

2 1

cat p

= − v v

Inte

rnal

loo

p –

Ther

mo

flu

id p

rob

lem

Explicit solution ofintermediate velocity

Pressure correctionPoisson equation withregularization

Velocity correction

Explicit time advance ofheat transport

2st

st st st st

p

Tc T

t

=

( )2

2alal al al al

p al

RTc T

T

t

I

S

= +

Implicit solution of heattransport in steel part

Implicit solution of heattransport and generationin aluminum part

Mai

n s

imu

lati

on

loo

p

RBF-FD discretization

2, , ,...i

Lu a Lx

= =

Problem

Shape functions ( )T 0.5 0.5u+

= =b W B W u Φu

Diff. operation ( )( ) ( )T 0.5 0.5ˆn

jjj

Lu L L u+

= = Φ

b W B W u

WLS ( )2

2 ˆn

j j j

j

r W u u= −

B – matrix of dimension (m x n)W – diagonal weight matrix

SVD / QR decomposition / NE - Cholesky decomposition

Moore Penrose pseudo inverse

( ) ( )1

0.5 0.5−

=α W B W u

Special case Interpolation =α Bu

1

ˆm

i i

i

u b=

=

Approximation coefficients

Basis functions (MQs, monomials, Gaussians, …)

Approximate solution over a

local support domain

Tu = b α

Implementation

Meshless(kNN + MLS)

Domain definitionvector<vec> nodes

Support sizesize_t n

Basis poolvector<function> b

Weight functionfunction W

ApproximationEngine

Position vectorDefines dimensionality of space

Type of basis functions

Type of weight function

WLS object

kNNEngine

Position vectorDefines dimensionality of space

WLS object

Refinement engine

Explicit operators

Implicit operators

Solution prototypes

Vec_t laplace(container<vec_t> &u, int node)

void laplace(matrix_t &M, int node, scalar_t alpha)

#pragma omp parallel for private(i) schedule(static)

for (i = 0; i < interior.size(); ++i) {

size_t c = interior[i];

///Heat transport

T2[c] = T1[c] + O.dt * op.lap(T1, c) - O.dt*op.grad(T1,c)*v1[c];

///Navier-Stokes

v2[c] = v1[c] + O.dt * ( - 1/O.rho * op.grad(P1,c).transpose()

+ O.mu/O.rho * op.lap(v1, c)

- op.grad(v1,c)*v1[c]);

///Mass continuity

scal_t dl2 = O.dl*O.dl; scal_t dt2 = O.dt*O.dt;

P2[c] = P1[c] - dl2*O.dt*O.rho * op.div(v1,c) + dt2*dl2 * op.lap(P1,c);

}

T1.swap(T2);

v1.swap(v2);

P1.swap(P2);

open source meshless projectMedusa: Coordinate Free Mehless Method implementation (MM)https://gitlab.com/e62Lab/medusa | http://e6.ijs.si/medusa/

Simulation of natural convection from Al490Fe65 conductor

10 CT =

.

40 CT = 80 CT =

Simulation of natural convection from Al490Fe65 conductor

.

Steady state temperature profile of air at different skin-ambient temperature differences.

Convergence plot (left) and time development at different skin-ambient temperature differences

Validation of numerical simulation

RB

F-FD –

nu

merical sim

ulatio

n, M

-m

easurem

ent

Result of Schlieren photography Simulated temperature

Experimental setup for measurements of conductor temperature

Experimental setup for Schlieren photography.

Conclusions

Improving DTR algorithms

We prepared a numerical simulation of heat transport within the overhead line andthermo-fluid transport in surrounding air with the ultimate goal to further improvetreatment of the most important cooling mechanism of overhead power line, i.e.convection.

The physical model is solved by an in-house implementation of RBF-FD method within theMedusa open source meshless project.

Model is validated by comparing simulation results, experimental data and IEEE and CIGREstandards.

Future work

Implement simulation of convective cooling in forced convection regime.

Prepare simulated relations for Nusselt number with respect to the wind velocity andgeometry of the power line.

Thank you for your attention