Calculating Electric Fields that drive Geomagnetically...
Transcript of Calculating Electric Fields that drive Geomagnetically...
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David Boteler, Ph.D.Natural Resources Canada
EPRI – NERC GIC and System Analysis WorkshopApril 18-20, 2012
Calculating Electric Fields that drive Geomagnetically Induced Currents
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Electric Field Calculation
Network Modelling
Displays
ElectrojetMagnetic Field
Earth Conductivity Models
System Information
USERS
This Talk
The Overall Process
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Contents
• Part 1. Nature of the Electric Fields
– Mutual Inductance
– Transfer Function
• Part 2. Calculation Methods
– a) Earth Conductivity Model
– b) Plane Wave Source
– c) Line Current Source
• Part 3. Electric Field Values
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• Mutual Inductance (E/I)
– Carson’s Equations
– Electrojet Source
• Transfer Function (E/B)
– Plane wave source
– High-pass filter
1. Nature of the Electric Fields
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1. Mutual Inductance
Carson’s Formulas
212
1=
E ZI 2 12 1=E Z I
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1. Mutual Inductance
Solutions for Carson’s Formulas
Starting point: Integral Equation
Series Expansion (Carson)
Series Expansion (Dubanton)
Complex Image Method (Deri et al)p
h+p
Dij
h
dij
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1. Mutual Inductance
Solutions for Carson’s Formulas
Starting point: Integral Equation
Series Expansion (Carson)
Series Expansion (Dubanton)
Complex Image Method (Deri et al)
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Inductive Coupling
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1. Mutual Inductance
Power System Calculations
p
h+pDij
h
dij
Source Current
height: 10 m
frequency: 60 Hz
Electrojet CalculationsSource Current
height: 100 km
frequency: 0.01 Hz
p
h+p Dij
hdij
E =IE = I I
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1. Mutual Inductance
Solutions for Electrojet Formulas
Starting point: Integral Equation (Price)
Series Expansion (Pirjola)
Complex Image Method (Boteler and Pirjola)
Apply to GIC Studies (Albertson and Van Baelen)
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Electric Field Calculation (1)Transfer Function
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Transfer Function
General Equation
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Transfer Function
Horizontal variation << vertical variation (due to skin depth effect)
X X
Often referred to as a “plane wave” source
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Transfer Function
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E(ω) = Z(ω) H(ω)
For a uniform earth, surface impedance Zi
=ωµσ
For a plane wave source:
Transfer Function
E(ω) H(ω) Z(ω)
ω
High Pass Filter
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2. Electric Field Calculations
• Earth Conductivity Models
– Skin Depth
– Rock Resistivities
– Calculate Earth Response
• Calculations for Plane Wave Source
• Calculations for Electrojet Source
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2a. Earth Conductivity Model
Skin Depthδ
ωµσ=
2
depth depth
Magnetic Field Magnetic Field
High frequency
Low frequencyHigh conductivity
Low conductivity
km
100skm
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2a. Earth Conductivity Model
Earth Structure Rock Resistivities
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Locations and Examples of 1-D Conductivity Models
2a. Earth Models
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Calculate Earth Response
μ – permeabilityω – frequencyZn – impedance in layer nσn – conductivity layer ndn – depth of layer nkn – propagation constant for layer n
NN k
iZ ωµ=Last layer:
ωµ
ωµ
iZk
iZk
rn
n
nn
n1
1
1
1
−
−
+
−= nn ik ωµσ=
Recurrence Relation
( )
n n
n n
2k dn
n 2k dn n
1 r eZ ik 1 r e
ωµ−
−
−= +
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Electric Field Calculation (1)
E(ω) = Z(ω)H(ω)
‘Plane Wave’ approximation: ∂H/∂x << ∂H/∂z
FFT IFFTMagnetic Field
Electric Field
FFTFFTFFT IFFTFFT IFFTFFT Electric Field
Magnetic Field
2b. Electric Field Calculation (Plane Wave)
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2b. Electric Field Calculation (Plane Wave)
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Frequency Domain & Time Domain
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Impulse Response
Time Domain Frequency Domain
Real
Imaginary
t
t
t
ω
ω
0 0
a)
b)
c)
Time Domain Frequency Domain
Real
Imaginary
t
t
t
ω
ω
0 0
a)
b)
c)
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Electric Field Calculation (1)Electric Field Calculation (Line Current)
• Approximate by a line current• Fields from external current from Biot-Savart law• Need to include effect of induced currents• Exact expression involves integration over all wavenumbers• Use Complex Image approximation
Electrojet Source
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Electric Field Calculation (1)Electric Field Calculation (Line Current)
Complex Image Method
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Electrojet Source
• Approximate by a line current• Fields from external current from Biot-Savart law• Need to include effect of induced currents• Exact expression involves integration over all
wavenumber• Use Complex Image approximation
• “Wide Electrojet” approximation
Electric Field Calculation (Line Current)
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Wide electrojet represented by a current density, j(x)with a Cauchy distribution of half-width a
Electric Field Calculation (Wide Electrojet)
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3. Electric Field Values
Effects of Different Parameters
Amplitude of Magnetic Disturbance
Frequency of Magnetic Field Variation
Amplitude and Frequency related
Earth Resistivity
Proximity to Source
Effective Resistivity and Frequency related
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3. Earth Conductivity Variations (Small Scale)
≈100 km
Ignore earth conductivity variations on scale less than length of transmission lines
Use E field averaged over length of line
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3. Earth Conductivity Variations (Large Scale)
Land Sea
Coast Effect
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Questions ?
Calculating Electric Fields