The need for accurate assessment of the geomagnetic hazard to power systems is driving a requirement to model geomagnetically induced currents (GIC) in multiple voltage levels of a power network. The Lehtinen–Pirjola method for modelling GIC is widely used but was developed when the main aim was
to model GIC in only the highest voltage level of a power network. Here we present a modification to the Lehtinen–Pirjola (LP) method designed to
provide an efficient method for modelling GIC in multiple voltage levels. The LP method calculates the GIC flow to ground from each node. However,
with a network involving multiple voltage levels, many of the nodes are ungrounded, i.e. have infinite resistance to ground, which is numerically inconvenient. The new modified Lehtinen–Pirjola (LPm) method replaces the earthing impedance matrix [

© Her Majesty the Queen in Right of Canada, as represented by the Minister of Natural Resources, 2022.

Geomagnetic disturbances produce geoelectric fields that drive geomagnetically induced currents (GIC) in power networks. These GIC flow along transmission lines and through transformer windings, where they can cause half-cycle saturation leading to harmonic generation, increased consumption of reactive power and transformer heating. These, in turn, can cause misoperation of protective relays and voltage sag and, in extreme cases, damage to transformers and system collapse (Kappenman and Albertson, 1990; Bolduc, 2002; Molinski, 2002; Kappenman, 2012; Guillon et al., 2016). A key requirement for understanding the impact of geomagnetic disturbances on power networks is the ability to model the GIC produced in a network by specified geoelectric fields. In 1985, Lehtinen and Pirjola published a landmark paper that provides the first description of a stand-alone method for modelling GIC. The Lehtinen–Pirjola (1985) method (hereafter referred to as the “LP method”) has been widely used in the geophysics and space weather community and provided the basis for GIC studies in many countries (e.g. Pirjola and Lehtinen, 1985; Mäkinen, 1993; Mäkinen et al., 1993; Thomson et al., 2005; Wik et al., 2008; Viljanen et al., 2012; Torta et al., 2014; 2017; Divett et al., 2018).

The LP method was designed at a time when mostly only the highest voltage levels of a power network were considered in GIC calculations. This was because the transmission lines at the lower voltage levels have higher resistance and so will experience smaller GIC values. However, in a desire to provide more comprehensive modelling of GIC in a power network, many modern studies are now looking to model GIC in multiple voltage levels in a power network. The LP method has been effectively used for such studies (e.g. Mäkinen, 1993; Mäkinen et al., 1993; Viljanen et al., 2012; Divett et al., 2018); however, using the LP method for multiple voltage levels involves many ungrounded nodes, thus having infinite resistance to ground, which is numerically inconvenient. Also, the main focus of the LP method was the GIC flow to ground through the transformer primary windings, which was the desired output when modelling a single voltage level of a power network. However, models for multiple voltage levels require calculation of the nodal voltages which are then used to calculate the GIC in the transformer windings (Boteler and Pirjola, 2014, 2017).

In this paper we show how the LP method can easily be modified to efficiently model GIC in multiple voltage levels of a power network by converting the LP method to calculate the nodal voltages directly. First we summarize the steps in the LP method and then show how these can be modified to give the modified Lehtinen–Pirjola method (hereafter referred to as the “LPm method”). We also show that the LPm method involves inversion of a matrix that is symmetric positive definite, allowing the use of efficient methods including sparse matrix techniques. Then we show how software for GIC calculations using the LP method can easily be converted to the LPm method and provide example calculations for the benchmark model introduced by Horton et al. (2012), including tables of values at intermediate steps, to help people transitioning their modelling from the LP method to the LPm method.

The GIC modelling method derived by Lehtinen and Pirjola (1985), the “LP” method, is produced by starting with Kirchhoff's current law that the net
current flowing into a node,

Substituting Eq. (2) into Eq. (1) gives (LP Eq. 9)

Note that, when considering multiple voltage levels, branches in the network consist of not just transmission lines, but also transformer windings. The transmission lines experience the driving emf produced by the magnetic field variations, whereas the transformer windings do not.

The driving emf in each transmission line is represented by an equivalent current source:

The equivalent current sources are then summed to give the current source directed into each node (LP Eq. 13).

Making this substitution in Eq. (3) gives

Thus,

The first summation represents the dependence of current

The second summation represents the dependence of current

This can be written in matrix form:

LP makes the substitution

Substituting Eq. (12) into Eq. (11) gives a matrix equation involving only the node to ground currents [

Gathering terms in [

The values of [

Now, when modelling the GIC in multiple voltage levels of a power network, many of the nodes are ungrounded. However, the LP method needs to specify
an earthing impedance for each node. This is done by adding “virtual” connections to ground from each ungrounded node (Mäkinen, 1993; Pirjola,
2005). These virtual earthing connections have infinite resistance, but this cannot be represented in the earthing impedance matrix
[

When modelling GIC in multiple voltage levels of a power network, it is necessary to calculate the nodal voltages before calculating the GIC in the transmission lines and transformer windings. In the LPm method the matrix equations are modified to provide a solution in terms of the nodal voltages. This also has the advantage that there is no need to add virtual earthing connections to ground from the ungrounded nodes.

To convert the currents flowing to ground [

The LP method allows for the [

Then Eq. (19) can be rewritten as

The voltages of the nodes are then found by taking the inverse of the sum of the admittance matrices and multiplying by the nodal current sources:

These node voltages can then be substituted into Eqs. (2) and (

The LPm method involves inversion of a matrix ([

GIC modelling is now being used, not just to assess the GIC for specified electric field values, but also to determine the variation of GIC throughout a geomagnetic disturbance. If the network configuration does not change during that time (not always the case), then the matrix inversion does not need to be recalculated at every time step.

If the electric field is assumed to be uniform across the network, then linear superposition can be used to calculate the GIC (Boteler, 2013). (A uniform electric field would be produced, e.g. if calculations are made using data from a single magnetic observatory and a one-dimensional (1-D) earth conductivity model.) The GIC modelling can be made for two cases: (i) a northward electric field of 1

This concept can be extended for using two magnetic observatories (Boteler et al., 2014), but this still requires use of a single 1-D earth conductivity model for the whole network.

In practice there is considerable variability in the earth conductivity structure across a power network. There are many modelling techniques for
calculating the electric fields in such cases, ranging from use of multiple 1-D earth models (Marti et al., 2014) to use of magnetotelluric transfer
functions and 3-D earth conductivity models (Weigel, 2017). In these cases, the electric fields across the network can change from place to place and
from one time step to the next. This will result in a different set of nodal current sources [

However, for GIC calculations using the LPm method, even more efficient time series calculations are possible. The solution of a matrix equation such as Eq. (21) can be accomplished using LU decomposition, as explained in Press et al. (2007). This involves writing the matrix
([

For a positive-definite symmetric matrix, as is obtained with the LPm method, the [

This Cholesky decomposition to solve the linear set is

The advantage is that the solution of a triangular set of equations is quite trivial, as Eq. (

To illustrate the differences between the LP method and the LPm method, consider the circuit for a substation with a two-winding transformer and three autotransformers as shown in Fig. 1. The LP method requires the addition of virtual connections to ground from nodes 1 and 2, as explained above. However, in the LPm method the connection to ground is expressed as an admittance value. For the ungrounded nodes the admittance to ground is zero, which can easily be included in the earthing admittance matrix without having to add virtual connections to the circuit.

Substation with a two-winding transformer and three autotransformers and the equivalent circuits for the LP and LPm methods.

The steps involved in calculating GIC in multiple voltage levels of a power network using the LP method and the LPm method are summarized in Fig. 2. In the LPm method, because it involves only admittances and calculates the nodal voltages directly, there is no need to add virtual connections to ungrounded nodes, and then there is no need to convert the currents through the virtual connections to nodal voltages.

Comparison of the steps involved in the LP and LPm methods.

Figure 2 also shows how easy it is to convert from the LP method to the LPm method. Many steps in the process are the same. The only changes are to
set up the earthing admittance matrix [

Single-line diagram of the benchmark test case of Horton et al. (2012).

To illustrate the use of the LPm technique, we present the calculation of GIC in the benchmark model of Horton et al. (2012) shown in Fig. 3. The following tables will also provide values for testing when converting software from the LP method to the LPm method.

To construct the network admittance matrix [

Assignment of node numbers.

Network admittance matrix [

The network admittance matrix [

The earthing admittance matrix [

Earthing admittance matrix [

In the earthing admittance matrix, most nodes are not connected to ground, so their earthing admittance values are zero, and there are only non-zero admittance values for the six neutral point nodes. Earthing admittance matrix values for the benchmark model are shown in Table 3.

Note that the general theory is expressed in terms of impedance and admittance, which can have reactive components, but, in practice, at the frequencies applicable to GIC the reactive components are negligible, and the network characteristics can be described as purely resistive or conductive.

Inverted matrix ([

The resulting inversion of the matrix gives ([

Horton et al. (2012) consider two cases: a northward electric field of 1

Voltages in the transmission lines and equivalent current sources for northward and eastward electric fields of 1

The voltage source in each transmission line and the equivalent current source, calculated from Eq. (

Nodal current sources for northward and eastward electric fields of 1

The nodal current sources are then combined with the inverted matrix (Eq. 22) to give the nodal voltages shown in Table 7. For nodes 1–11 these give the bus voltages shown in Table V of Horton et al. (2012). For nodes 12–18, combining the nodal voltage and the substation grounding resistance gives the GIC flow to ground for each substation shown in Table VII of Horton et al. (2012).

Nodal voltages produced by northward and eastward electric fields of 1

The nodal voltages substituted into Eqs. (2) and (

The above example calculation shows that the LPm method provides GIC values that exactly match those for the benchmark model provided by Horton et al. (2012). The values in Tables 2 to 7 can also be used to check intermediate steps in the software used for the LPm calculations.

Any software developed to model GIC should be able to exactly match the values provided by the Horton et al. (2012) paper. The results presented in that paper are not an average of modelling results nor an approximation to the correct values but are the identical values obtained using four different software implementations. However, initial calculations involving the four different software implementations provided similar but slightly different results. Further investigation showed that the origin of the differences was in the way that distances between substations were being calculated in the different implementations. Some versions used formulas based on a spherical earth and some used formulas taking account of its non-spherical shape. It was then decided to standardize on substation latitudes and longitudes based on the WGS84 ellipsoid model of the Earth which is used by the global navigation satellite system (GNSS) for geolocation. After this, all the calculations gave exactly the same results. To get the source voltage values presented in Table 5 (and hence match the GIC results for the benchmark model) thus requires using the formulas presented in the Appendix of Horton et al. (2012) for calculating distances between substations.

Many people have used the LP method for calculating GIC in the highest voltage level of their power networks. With the increasing requirement to
calculate GIC in multiple voltage levels of a power network, it is hoped that the new LPm method described above will provide an easy way to convert existing LP software. The conversion is a simple process. Just replace the earthing impedance matrix [

Power networks, on average, have three transmission lines and one or two transformer windings connected to a bus, so a typical row in the admittance
matrix has only five or six non-zero elements, independent of the overall network size. Thus, for larger networks, where node numbers can be in the thousands, the admittance matrix will have over 99 % of its values equal to zero. Cholesky factorization takes advantage of this fact by making
use of sparse matrix methods (Stott and Alsaç, 1987; Press et al., 2007), thus additionally reducing memory usage and computation time. To examine
how this affects the GIC modelling, we performed calculations for two networks using both the LP and LPm methods. The networks modelled were (1) the benchmark network of Horton et al. (2012), which has 18 nodes, and (2) the nation-wide Spanish Power Grid operated by Red Eléctrica de
España (REE), which has 1388 nodes. GIC in the 400

Properties of the matrices to be inverted using the LP and LPm methods for different power networks, namely the Horton et al. (2012) benchmark and REE.

Tests we did showed that the LP and LPm methods both produce matrices that are sparse, so there is potential for sparse matrix techniques to be
applicable. Table 8 shows the calculation times and memory usage for GIC calculations using the LP and LPm methods. These show that memory usage was
drastically reduced when using sparse matrix techniques, with the reduction being more significant with the larger REE network. The time for the
matrix inversion is significantly affected, as expected, by the size of the matrix involved. For the Horton network (18

The “Inversion time” column in Table 8 reflects the time required to compute

We have presented a new version of the LP method, modified for efficient modelling of GIC in multiple voltage levels of a power system. In the LPm
method the earthing impedance matrix, [

Multiplication of the inverted matrix ([

Guidance is provided for converting software from the LP method to the LPm method and an example calculation using the benchmark model of Horton et al. (2012) is presented to provide a set of values for testing GIC calculation software.

Calculations of GIC using the LPm method involve a matrix that is symmetric positive definite. This enables a solution to be obtained by Cholesky decomposition, (a specific case of LU decomposition), which is numerically more accurate than computing the matrix inversion itself. The factorization of Cholesky decomposition can always be implemented using sparse matrix techniques, speeding up the calculations for large networks.

Thus the LPm method provides an efficient method for calculating GIC in multiple voltage levels in a power network that provides a valuable tool for assessing the geomagnetic hazard to power systems.

No datasets were used in this research.

RJP and DHB planned the work and developed the theory. LT developed the modelling code and generated the example results. SM performed the tests of matrix inversion. RJP and DHB wrote the manuscript draft. LT and SM reviewed and edited the manuscript.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was performed as part of the Public Safety Geoscience program and Canadian Hazards Information Service of Natural Resources Canada. Santiago Marsal would like to thank Red Eléctrica de Españ˜a (REE) for supporting this study. Natural Resources Canada contribution number 20210276.

This work was supported by Natural Resources Canada. Santiago Marsal was supported by the projects (grant no. CGL2017-82169-C2-1-R) and (grant no. PID2020-113135RB-C32) both funded by FEDER/Ministerio de Ciencia, Innovación y Universidades – Agencia Estatal de Investigación. His research that led to these results was also carried out partly with funds from “La Caixa” foundation.

This paper was edited by Georgios Balasis and reviewed by Ciaran Beggan and one anonymous referee.