Calculating Electric Fields that drive Geomagnetically Induced Currents

Transcription

1 Calculating Electric Fields that drive Geomagnetically Induced Currents David Boteler, Ph.D. Natural Resources Canada EPRI NERC GIC and System Analysis Workshop April 18-20, 2012

2 The Overall Process Magnetic Field Electrojet This Talk Earth Conductivity Models Electric Field Calculation Network Modelling Displays System Information U S E R S 2

3 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 3

4 1. Nature of the Electric Fields Mutual Inductance (E/I) Carson s Equations Electrojet Source Transfer Function (E/B) Plane wave source High-pass filter 4

5 1. Mutual Inductance Carson s Formulas E I 2 1 = Z 12 E2 = Z12I1 5

6 1. Mutual Inductance Solutions for Carson s Formulas Starting point: Integral Equation Series Expansion (Carson) Series Expansion (Dubanton) Complex Image Method (Deri et al) h p d ij h+p D ij 6

7 1. Mutual Inductance Solutions for Carson s Formulas Starting point: Integral Equation Series Expansion (Carson) Series Expansion (Dubanton) Complex Image Method (Deri et al) 7

8 Inductive Coupling 8

9 1. Mutual Inductance Power System Calculations Source Current height: 10 m frequency: 60 Hz Electrojet Calculations Source Current height: 100 km frequency: 0.01 Hz d ij h h d ij p p h+p D ij h+p D ij E = I I I 9

10 1. Mutual Inductance Solutions for Electrojet Formulas Starting point: Integral Equation (Price) Apply to GIC Studies (Albertson and Van Baelen) Series Expansion (Pirjola) Complex Image Method (Boteler and Pirjola) 10

11 Electric Field Calculation (1) Transfer Function 11

12 Transfer Function General Equation 12

13 Transfer Function X X Horizontal variation << vertical variation (due to skin depth effect) Often referred to as a plane wave source 13

14 Transfer Function 14

15 Transfer Function For a plane wave source: E(ω) = Z(ω) H(ω) For a uniform earth, surface impedance Z = iωµ σ High Pass Filter H(ω) Z(ω) E(ω) ω 15

16 2. Electric Field Calculations Earth Conductivity Models Skin Depth Rock Resistivities Calculate Earth Response Calculations for Plane Wave Source Calculations for Electrojet Source 16

17 2a. Earth Conductivity Model Skin Depth δ = 2 ωµσ Magnetic Field Magnetic Field km depth High frequency depth 100s km High conductivity Low frequency Low conductivity 17

18 2a. Earth Conductivity Model Earth Structure Rock Resistivities 18

19 2a. Earth Models Locations and Examples of 1-D Conductivity Models 19

20 Calculate Earth Response Recurrence Relation Z n 2kndn 1 re n = iωµ 2knd n k ( 1+ re ) n n μ permeability ω frequency Z n impedance in layer n σ n conductivity layer n d n depth of layer n k n propagation constant for layer n r n 1 1+ k k n n Last layer: Z n 1 iωµ Z n 1 iωµ = n n Z N = k = iωµ k N iωµσ 20

21 Electric Field Calculation (1) 2b. Electric Field Calculation (Plane Wave) Plane Wave approximation: H/ x << H/ z Magnetic Field FFT E(ω) = Z(ω)H(ω) IFFT Electric Field 21

22 2b. Electric Field Calculation (Plane Wave) 22

23 Frequency Domain & Time Domain 23

24 Impulse Response Time Domain Frequency Domain a) 0 t 0 ω b) t ω c) Real Imaginary t 24

25 Electric Field Calculation (1) Electric Field Calculation (Line Current) 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 wavenumbers Use Complex Image approximation 25

26 Electric Field Calculation (1) Electric Field Calculation (Line Current) Complex Image Method 26

27 Electric Field Calculation (Line Current) 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 27

28 Electric Field Calculation (Wide Electrojet) Wide electrojet represented by a current density, j(x) with a Cauchy distribution of half-width a 28

29 3. Electric Field Values Effects of Different Parameters Amplitude of Magnetic Disturbance Frequency of Magnetic Field Variation Earth Resistivity Amplitude and Frequency related Effective Resistivity and Frequency related Proximity to Source 29

30 3. Earth Conductivity Variations (Small Scale) 100 km Use E field averaged over length of line Ignore earth conductivity variations on scale less than length of transmission lines 30

31 3. Earth Conductivity Variations (Large Scale) Coast Effect Land Sea 31

32 Calculating Electric Fields Questions? 32