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