Calculating Electric Fields that drive Geomagnetically...
Transcript of Calculating Electric Fields that drive Geomagnetically...
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
2© 2012 Electric Power Research Institute, Inc. All rights reserved.
Electric Field Calculation
Network Modelling
Displays
ElectrojetMagnetic Field
Earth Conductivity Models
System Information
USERS
This Talk
The Overall Process
3© 2012 Electric Power Research Institute, Inc. All rights reserved.
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
4© 2012 Electric Power Research Institute, Inc. All rights reserved.
• 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
5© 2012 Electric Power Research Institute, Inc. All rights reserved.
1. Mutual Inductance
Carson’s Formulas
212
1=
E ZI 2 12 1=E Z I
6© 2012 Electric Power Research Institute, Inc. All rights reserved.
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
7© 2012 Electric Power Research Institute, Inc. All rights reserved.
1. Mutual Inductance
Solutions for Carson’s Formulas
Starting point: Integral Equation
Series Expansion (Carson)
Series Expansion (Dubanton)
Complex Image Method (Deri et al)
8© 2012 Electric Power Research Institute, Inc. All rights reserved.
Inductive Coupling
9© 2012 Electric Power Research Institute, Inc. All rights reserved.
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
10© 2012 Electric Power Research Institute, Inc. All rights reserved.
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)
11© 2012 Electric Power Research Institute, Inc. All rights reserved.
Electric Field Calculation (1)Transfer Function
12© 2012 Electric Power Research Institute, Inc. All rights reserved.
Transfer Function
General Equation
13© 2012 Electric Power Research Institute, Inc. All rights reserved.
Transfer Function
Horizontal variation << vertical variation (due to skin depth effect)
X X
Often referred to as a “plane wave” source
14© 2012 Electric Power Research Institute, Inc. All rights reserved.
Transfer Function
15© 2012 Electric Power Research Institute, Inc. All rights reserved.
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
16© 2012 Electric Power Research Institute, Inc. All rights reserved.
2. Electric Field Calculations
• Earth Conductivity Models
– Skin Depth
– Rock Resistivities
– Calculate Earth Response
• Calculations for Plane Wave Source
• Calculations for Electrojet Source
17© 2012 Electric Power Research Institute, Inc. All rights reserved.
2a. Earth Conductivity Model
Skin Depthδ
ωµσ=
2
depth depth
Magnetic Field Magnetic Field
High frequency
Low frequencyHigh conductivity
Low conductivity
km
100skm
18© 2012 Electric Power Research Institute, Inc. All rights reserved.
2a. Earth Conductivity Model
Earth Structure Rock Resistivities
19© 2012 Electric Power Research Institute, Inc. All rights reserved.
Locations and Examples of 1-D Conductivity Models
2a. Earth Models
20© 2012 Electric Power Research Institute, Inc. All rights reserved.
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−
−
−= +
21© 2012 Electric Power Research Institute, Inc. All rights reserved.
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)
22© 2012 Electric Power Research Institute, Inc. All rights reserved.
2b. Electric Field Calculation (Plane Wave)
23© 2012 Electric Power Research Institute, Inc. All rights reserved.
Frequency Domain & Time Domain
24© 2012 Electric Power Research Institute, Inc. All rights reserved.
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)
25© 2012 Electric Power Research Institute, Inc. All rights reserved.
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
26© 2012 Electric Power Research Institute, Inc. All rights reserved.
Electric Field Calculation (1)Electric Field Calculation (Line Current)
Complex Image Method
27© 2012 Electric Power Research Institute, Inc. All rights reserved.
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)
28© 2012 Electric Power Research Institute, Inc. All rights reserved.
Wide electrojet represented by a current density, j(x)with a Cauchy distribution of half-width a
Electric Field Calculation (Wide Electrojet)
29© 2012 Electric Power Research Institute, Inc. All rights reserved.
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
30© 2012 Electric Power Research Institute, Inc. All rights reserved.
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
31© 2012 Electric Power Research Institute, Inc. All rights reserved.
3. Earth Conductivity Variations (Large Scale)
Land Sea
Coast Effect
32© 2012 Electric Power Research Institute, Inc. All rights reserved.
Questions ?
Calculating Electric Fields