Transcript Statistical model of wind farms for power flow

FINITE ELEMENTS SOFTWARE
FOR ELECTROMAGNETICS
APPLIED TO ELECTRICAL
ENGINEERING TRAINING.
J. Mur, J.S. Artal, J. Letosa, A. Usón
and M. Samplón
Electrical Engineering Department
Introduction
Electromagnetics is difficult to learn by the
student
 Show practical engineering examples related with the
Maxwell equations
 Simplify the problem to avoid high-level mathematics
But simplifications can lead to errors in
industrial applications…
 Presentation of Electromagnetics theory in a more
engaging manner, using a breakthrough technology as a
bridge between the classical theoretical approach and real
engineering applications
Strategy to introduce the numerical
simulation to the students (I)
Study case: resistor with complex geometry
solved with three different techniques
 Two of them are analytical solutions of simplified
models of the resistor which can be easily solved
by the student
 In the third one, the resistor was modelled and
numerically solved using a commercial FES
(Vector Fields)
 Graphics from FES help to understand the effect of
small curvature radius in the inhomogeneous
distribution of Joule power dissipation.
Strategy to introduce the numerical
simulation to the students (II)
The numerical solution is compared with
the results from the analytical methods.
 The student realize that suitable values can be
obtained by analytic solutions, but better precision
require sophisticated tools.
Real photographs of temperature
distribution during lab tests of a small
automotive component.
 The student feels not being solving another academic
problem.
The simplified resistor:
S-shape conductor sheet, with a negligible
thickness, which can be considered to have only
two dimensions :
c
V0
a
b
Conductivity 
thickness h
c
V0
c = a = 11 mm
b = 4 mm
 = 1,515 10-3  m
h = 0.1 mm
V0 = 15 V
First approach (analytical)
Compute the resistance as if it were a
cable:



Compute mean length
Compute mean cross section
Apply the formula of resistance of a cable
(Cartesian symmetry)
R
lmean
2  rmean  2 c


Ssection
( a - b) h
2
ab
2 c
2
( a - b) h
Second approach (analytical) I
Split the piece in four independent
resistors which can be connected in
series :
R2
R1
R2
R1
R1
R2
R1
R2
Second approach (II)
The resistance of each zone can be
easily computed independently:
Straight zone
l
c
R   mean  
Ssection
( a  b) h
Resistance, R
Current
J 
density, J
Power density
p  J ·E   J
2
p
I
Ssection
I2
Ssection
2

I
(a  b) h
I2

(a  b) 2 h 2
½ Circle
R
J 
p

 b
h ln a
I
 b
r h ln a
I2
 
r 2 h 2  ln a 
b 

2
Third approach (numerical) I
Numerical calculation of the electrical field
by means of the finite element method.
 The visual graphs obtained help the student to discover
some special features, that weren’t considered in the
analytical solutions.
Current density
distribution at
the simplified
piece:
Comparison of results for the
simplified case
While the first
 approximation predicts a constant
value for | J | and p in any point of the piece, the
second one expects values which are dependant
of the distance to the centre.
The results of the second method are closer to
the ones obtained by FEM, both fortotal values
as resistance and for local values as J and p. The
error for the total resistance in the first method
with respect to the FEM value is 4,3% and 1,2%
in the second one.
Note: the FEM is only used as a tool and its foundations
are not explained to the students of a basic course.
Sources of inaccuracy in the two
analytical methods (I)

Lines of J show a transition in the zones of
change of change of curvature.
 = /2

J
=
junction
at  = 0
In the analytical
solutions, this
feature is not
considered => the
continuity law is
not accomplished
Sources of inaccuracy in the two
analytical methods (II)
The redistribution of current in the joins
also lead to changes in equipotential lines.

Graph of potential around the centre of the resistance
Analysis of a real-case resistance
The procedure to calculate the electric field
and the electrical resistance is extended to
cope with an industrial resistor.
 As the students have already understand the
resolution procedure for a simplified case, they can
focus in the special features of this case.
Distribution of temperature of the
resistance
A high electrical current was applied to the resistor
to heat it below 973 K. Colour of resistance is
compared with the FEM solution.

Radiation at that temperature becomes visible, and
colour in each point of the surface gives an indication of
the different temperature.
Conclusions
Introduction of realistic problem-solving
techniques, combining basic mathematics
theory and separation into elemental
straightforward cases, stressing the physics
behind.
The accuracy of the solutions depends on
several factors (temperature, skin effect...)
Questions?