Mathematical Problems in Engineering

Volume 2015, Article ID 761964, 10 pages

http://dx.doi.org/10.1155/2015/761964

## An Approach to Model Earth Conductivity Structures with Lateral Changes for Calculating Induced Currents and Geoelectric Fields during Geomagnetic Disturbances

^{1}Beijing Key Laboratory of High Voltage and EMC, North China Electric Power University, Beijing 102206, China^{2}Finnish Meteorological Institute, 00101 Helsinki, Finland^{3}Natural Resources Canada, Ottawa, ON, Canada K1A 0Y3^{4}State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China

Received 30 August 2014; Revised 22 January 2015; Accepted 22 January 2015

Academic Editor: Hari M. Srivastava

Copyright © 2015 Bo Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

During geomagnetic disturbances, the telluric currents which are driven by the induced electric fields will flow in conductive Earth. An approach to model the Earth conductivity structures with lateral conductivity changes for calculating geoelectric fields is presented in this paper. Numerical results, which are obtained by the Finite Element Method (FEM) with a planar grid in two-dimensional modelling and a solid grid in three-dimensional modelling, are compared, and the flow of induced telluric currents in different conductivity regions is demonstrated. Then a three-dimensional conductivity structure is modelled and the induced currents in different depths and the geoelectric field at the Earth’s surface are shown. The geovoltages by integrating the geoelectric field along specific paths can be obtained, which are very important regarding calculations of geomagnetically induced currents (GIC) in ground-based technical networks, such as power systems.

#### 1. Introduction

During geomagnetic disturbances (GMD), geomagnetically induced currents (GIC) driven by the geoelectric field are flowing in ground-based electrical systems such as electric power transmission networks with neutral grounded transformers. The geoelectric field is the key factor for determining the GIC in power systems, and it can be determined when the sources of the GMD and the Earth conductivity structure are known. For now, it has been usually assumed that the geoelectric field is spatially uniform, which enables a simple calculation of voltages in the transmission lines that drive GIC in the network and through the transformers, for example, [1, 2]. Unfortunately, in the real world, the complexities of ionospheric-magnetospheric source current systems and the inhomogeneities in the Earth’s conductivity structure make the geoelectric field nonuniform [3]. Therefore, the questions arise of how to model and calculate the geoelectric field more accurately and what the exact effects of lateral conductivity variations are on geoelectric fields, on currents flowing within the Earth (called telluric currents), and on GIC.

In geophysical research area, many techniques are used for determining the geoelectric fields and the methods assume different distributions of the source currents and different conductivity structures [4]. As pointed out in [5], here we need to emphasize again that, in electrical engineering research, evaluation of the impacts of GMD on power systems is the primary interest, which is different from the main focus of geophysicists, who mainly want to infer the conductivity profile of the Earth’s interior. Anyway, forward modelling of geomagnetic induction in the Earth introduces many numerical methods and modelling techniques which are good references for research of GMD effects on power systems as well [6].

The methods and techniques to calculate the geoelectric field depend on the model of the Earth conductivity structure and on the GMD source approximation. A widely used model of the Earth is a one-dimensional (1D) conductivity structure, where the Earth is described as a homogeneous or layered semi-infinite conductor with given conductivity values in each layer. The air region is set on the top of the Earth’s surface with the conductivity value equal to zero. The source of GMD is assumed to be a vertically incident plane wave or a line current located at a certain height. In the plane wave case, the geoelectric field occurring at the Earth’s surface can be determined by multiplying the geomagnetic variation by the surface impedance, which is a recursive function including the depths and conductivity values of different layers [7]. This process is known as the “plane wave method” [8]. This technique is invalid if the source differs much from a plane wave [9], which is the situation, for example, for a line current source representing an ionospheric electrojet. In this case, some other techniques, such as the complex image method (CIM) [10] or the Fast Hankel Transform (FHT) method [11], can be used to calculate the electric and magnetic fields at the surface of the Earth. The key point of CIM is that the telluric currents can be approximated by an image of the ionospheric source current located in a complex space, thus enabling closed-form expressions of the electric and magnetic fields at the Earth’s surface. FHT allows fast computations of the integrals involved in the exact expressions of the fields. These approaches bring a lot of results on theoretical understanding of the ground effects of GMD [12].

If the Earth’s conductivity structure is 1D, sophisticated numerical techniques seem to be unnecessary for solving the electromagnetic induction problem because the methods mentioned above can determine geoelectric fields fast and accurately enough. However, the realistic Earth structure is much more complicated than a 1D layered model. One typical structure contains the land-ocean interface and was analyzed in [5, 13, 14]. Although the “coast effect,” which is that the conductivity contrast between land and ocean will enhance the electric field at the landside close to the coastline, has been demonstrated before, simplified assumptions of the ionospheric source make the studies limited. On the other hand, the complicated processes occurring in the near-Earth space during GMD mean that it is impossible to derive detailed functions or formulae for the sources of geomagnetic variations.

In this paper, a typical Earth conductivity structure with lateral variations is modelled for calculating geoelectric fields based on the Finite Element Method (FEM). Due to the features of this structure, two-dimensional (2D) and three-dimensional (3D) implementations are introduced separately. Firstly, the governing equations and boundary conditions in both cases are explained briefly and the numerical results are compared. Then a more complicated Earth conductivity structure which can only be analyzed in 3D is modelled. Some applications and limitations of this approach are discussed at the end.

#### 2. Models and Methods

##### 2.1. Theoretical Background

In electromagnetic calculations related to GMD the displacement currents can be neglected due to the low frequencies involved [15, 16], so calculating the telluric currents and the geoelectric fields becomes a quasistatic field problem. In GMD research, the conductivity structure is generally assumed to be isotropic and the conductivity will have different values in different regions, but in each region the conductivity is assumed to be uniform. The permeability always has its free space value H/m.

Consider the Earth’s surface as a planar interface between Earth and air. To calculate the geoelectric field, the traditional computational model contains the Earth conductivity region and the air region containing a specific source. Since only the fields at the Earth’s surface and within the Earth are relevant for assessing GIC impacts, the Earth conductivity structure can be seen as a closed region where the Earth’s surface becomes its boundary. Based on the uniqueness of the solution of a boundary value problem [17], only the tangential component of geomagnetic field obtained from geomagnetic observatories in the real world [18] is necessary for determining the electromagnetic fields in the Earth. Under this circumstance, the air region and the external source need not be modelled, which reduces the computation.

To evaluate the approach presented above, a complex conductivity structure with lateral conductivity variations is considered, as shown in Figure 1. A Cartesian coordinate system is used, where points northwards, points eastwards, and points downwards. The Earth’s surface is set at . The conductivity structure under the Earth’s surface is assembled by two different one-layered models where the Quebec layered model is assumed for and the British Columbia (BC) layered model is assumed for [19]. The Quebec and BC models are typical for a resistive and a conducting ground, respectively. In our calculation, the geomagnetic field at the Earth’s surface is assumed to be uniform and to only have the component. These assumptions make all the field quantities have zero derivatives with respect to the coordinate. Therefore, the Galerkin FEM technique with both 2D planar elements and 3D solid elements can be used to solve this boundary value problem.