EM modeling and power cables
Sven Nordebo
Department of Physics and Electrical Engineering
Linnæus University, Sweden
2nd SSF workshop on Optimal and Automated Design of Electromagnetic Structures, Lund, 14 January, 2015.
Outline
Research activities
Electromagnetic losses in three-phase power cables
On the natural modes of helical structures
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 2(14)
Previous and planned research activities
Previous research activities
I
VR-ABB-PhD: Electromagnetic dispersion modeling and analysis for
power cables. Broadband pulse propagation on HVDC power cables,
for fault localization, length estimation, etc.
Current and planned research activities within the SSF-project: Complex
analysis and convex optimization for EM design
I
SSF-ABB-PhD: Electromagnetic losses in cable armour.
Characterization, measurements and estimation of the induced
conduction and hysteresis losses in the cable steel armour.
I
SSF-ABB-mobility: Electromagnetic losses in three-phase power
cables. On the natural modes of helical structures.
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 3(14)
Outline
Research activities
Electromagnetic losses in three-phase power cables
On the natural modes of helical structures
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 4(14)
Electromagnetic losses in three-phase power cables
There is a great potential of accurate electromagnetic modeling, analysis,
and optimization (design) of three-phase power cables, with regard to
I
conductor losses, skin-effect
I
induced conduction losses in metal sheaths and armour
I
iron losses in armour (hysteresis losses)
HVAC Cable Design: Submarine Three-Core
Metal sheath
Armour loss measurement: Thre
© ABB Group
April 24, 2013 | Slide 11
Inner plastic sheath
Conductor
Conductor shield
Insulation
Profiles
Insulation shield
Rated voltage
Max. system voltage
Conductor
Type/material
Cross-section
Diameter
Insulation system
Material 1
Material 2
Material 3
Metallic sheath
Type/material
Thickness
Diameter
Inner sheath
Type/material
Diameter
Optical cable
Cable assembly
Bedding
Type/material
Armouring
Type
Material 1
Material 2
Wire diameter
Diameter
Outer cover
Material
Complete cable
Diameter
Armour
Outer sheath (Polypropylene yarn)
Optical fiber
solid copper
240 mm²
17,5 mm
115 kV
123 kV
semi-conductive PE
XLPE
semi-conductive PE
lead alloy
2 mm
56 mm
PE
60 mm
polymeric tapes
Double armour
Galvanized steel (72+84
wires)
Bitumen
5 mm
151 mm
159 mm
Polypropylene yarn
Images from: Workshop on Mathematical Modelling of Wave Phenomena 2013, Linnæus University.
© ABB Group
April 24, 2013 | Slide 7
Presentation by Danijela Palmgren: http://lnu.se/polopoly_fs/1.85931!Palmgren.pdf
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 5(14)
Electromagnetic losses in three-phase power cables
Armour loss measurement: Armoured cable
© ABB Group
April 24, 2013 | Slide 13
Images from: Workshop on Mathematical Modelling of Wave Phenomena 2013, Linnæus University.
Presentation by Danijela Palmgren: http://lnu.se/polopoly_fs/1.85931!Palmgren.pdf
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 6(14)
Estimation of electromagnetic losses in cable armour
I
Build and model a transfomer based on the cable steel
I
Conduction losses (eddy currents) J = σE ⇒ D = 0 (r + i ωσ0 )E
I
Iron losses (hysteresis effect) B = µ0 µ(H)
I
How to estimate, simultaneously, the conduction losses as well as
the (non-linear) hysteresis losses in the cable steel?
I Linear model, proximity effects
I Micromagnetic models: Jiles-Atherton model (1984-1986)
I Variational models based on thermodynamics (1997-2011)
I
Identify model (material) parameters based on measurements.
EM-theory, estimation theory, convex optimization.
σ, µ
i
+
u
−
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 7(14)
Estimation of electromagnetic losses in cable armour
Modeling of magnetic hysteresis losses in steel
Left: Classical non-linear Jiles-Atherton model. Right: Linear model.
Hysteresis plot B [T] versus H [A/m]
2
Hysteresis plot B [T] versus H [A/m]
4
1
2
0
0
−1
−2
−2
−5,000
0
H [A/m]
5,000
−4
−5,000
0
H [A/m]
5,000
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 8(14)
Estimation of electromagnetic losses in cable armour
Modeling of magnetic hysteresis losses in steel
Left: Classical non-linear Jiles-Atherton model. Right: Linear model.
Hysteresis plot B [T] versus H [A/m]
0.5
Hysteresis plot B [T] versus H [A/m]
1
0.5
0
0
−0.5
−0.5
−1,000
0
H [A/m]
1,000
−1
−1,000
0
H [A/m]
1,000
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 9(14)
Outline
Research activities
Electromagnetic losses in three-phase power cables
On the natural modes of helical structures
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 10(14)
On the natural modes of helical structures
Floquet wave number for periodic structures J (r) = Je(r)eiβz
Periodic electric Green’s dyadic
Z Z p
E(r) = iωµ0 µ
Gep (r, r 0 , k, β) · J (r 0 ) dS 0 dz 0
S
0
Poisson summation formula
Gep (r, r 0 , k, β) =
∞
X
Ge (r, r 0 + ẑnp, k)eiβpn
n=−∞
=
∞
0
2π
1 X
2π
Ge (ρ, ρ0 , k, β + n )ei(β+n p )(z−z )
p n=−∞
p
Spectral representation of the free-space electric Green’s dyadic
Z ∞
0
0
1
eik|r−r |
1
Ge (r, r 0 , k) = {I+ 2 ∇∇}
=
Ge (ρ, ρ0 , k, α)eiα(z−z ) dα
0
k
4π|r − r |
2π −∞
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 11(14)
On the natural modes of helical structures
I
Periodic volume integral equation for the electric field,
based on the Floquet mode eiβz
Z Z p
[I + ρ̂ρ̂ · χ(r)]·E(r)−k 2
G0ep (r, r 0 , k, β)·χ(r 0 )·E(r 0 ) dS 0 dz 0 = 0
S
0
I
Expansion of the periodic dyadic Green’s function G0ep (r, r 0 , k, β) in
cylindrical vector waves.
I
A two-dimensional Fourier series expansion yields an infinite system
of coupled one-dimensional integral equations
Emn (ρ) + ρ̂ρ̂ ·
XX
m0
2
Z
−k 2π
0
χm−m0 ,n−n0 (ρ) · Em0 n0 (ρ)
n0
a
am (ρ, ρ0 , k, β+n
2π X X
χm−m0 ,n−n0 (ρ0 )·Em0 n0 (ρ0 )ρ0 dρ0
)·
p
0
0
m
n
=0
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 12(14)
On the natural modes of helical structures
Material twist-modes χmn in Fourier space
n
m
Azimuthal Floquet mode Emn containing the “excitation” factor eiφ
Quasi-static assumption: The mode shapes of E is the same as for χ
n
m
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 13(14)
On the natural modes of helical structures
Issues
I
Proper treatment of the source point (here in cylindrical
coordinates)
0
Ge (r, r 0 , k) = {I+
1
1
eik|r−r |
= G0e (r, r 0 , k)− 2 ρ̂ρ̂δ(r−r 0 )
∇∇}
2
0
k
4π|r − r |
k
I
The volume integral formulation is known to be “strongly” singular.
I
Each one-dimensional integral operator is only weakly singular
(discontinuous kernel).
I
Spectral theory, characterization of spectral properties similar to
Fredholm theory?
I
Analytic function theory view. Existence of modes (residues),
radiation modes are the contributions from an integration along the
branch-cut.
Sven Nordebo, Department of Physics and Electrical Engineering, Linnæus University, Sweden. 14(14)