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
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