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


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


• 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


1. Mutual Inductance

Carson’s Formulas

212

1=

E ZI 2 12 1=E Z I



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



Solutions for Carson’s Formulas

Starting point: Integral Equation

Series Expansion (Carson)

Series Expansion (Dubanton)

Complex Image Method (Deri et al)


Inductive Coupling



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



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)


Electric Field Calculation (1)Transfer Function


Transfer Function

General Equation


Transfer Function

Horizontal variation << vertical variation (due to skin depth effect)

X X

Often referred to as a “plane wave” source


Transfer Function


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


2. Electric Field Calculations

• Earth Conductivity Models

– Skin Depth

– Rock Resistivities

– Calculate Earth Response

• Calculations for Plane Wave Source

• Calculations for Electrojet Source


2a. Earth Conductivity Model

Skin Depthδ

ωµσ=

2

depth depth

Magnetic Field Magnetic Field

High frequency

Low frequencyHigh conductivity

Low conductivity

km

100skm


2a. Earth Conductivity Model

Earth Structure Rock Resistivities


Locations and Examples of 1-D Conductivity Models

2a. Earth Models


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−

−

−= +


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)


2b. Electric Field Calculation (Plane Wave)


Frequency Domain & Time Domain


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)


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


Electric Field Calculation (1)Electric Field Calculation (Line Current)

Complex Image Method


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)


Wide electrojet represented by a current density, j(x)with a Cauchy distribution of half-width a

Electric Field Calculation (Wide Electrojet)


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


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


3. Earth Conductivity Variations (Large Scale)

Land Sea

Coast Effect


Questions ?

Calculating Electric Fields

calculating electric fields that drive geomagnetically...

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