Eigenvalues and eigenvectors of the transfer matrix involved in the calculation of geomagnetically induced currents in an electric power transmission network
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2011
Received: 21 February 2011
Accepted: 9 June 2011
Published: 26 January 2012
Geomagnetically induced currents (GIC) flowing in power grids can be calculated from matrix equations whose input data are the geoelectric field and the network parameters. The transfer matrix between the “perfect-earthing” (pe) currents and the earthing GIC are discussed in this paper by considering its eigenvalues and eigenvectors. The pe currents include the influence of the geoelectric field whereas the transfer matrix only depends on the network data. It is shown that an eigenvalue equals one or the corresponding eigenvector satisfies the condition that the sum of its pe currents is zero. Using physical arguments, we conclude that all eigenvalues of the transfer matrix are non-negative and between zero and one. This statement is proved mathematically for a three-node network and supported by numerical computations for the Finnish 400 kV GIC test model. Special attention is paid to the norm of the earthing GIC, which gives an idea of the risk of GIC to a power grid. This norm seems to have the lower and upper limits practically equal to the smallest and largest (≠1) eigenvalue of the transfer matrix multiplied by the norm of the pe currents.
Geomagnetically induced currents (GIC) are ground effects of space weather, whose origin is in solar activity. When flowing in electric power transmission networks, oil and gas pipelines, telecommunication cables or railway circuits, GIC may cause problems to the particular system. The history of GIC dates back to the first telegraph systems in the mid-1800’s (e.g. Boteler et al., 1998; Lanzerotti et al., 1999). In power grids, the problems result from transformer saturation due to GIC (e.g. Bolduc, 2002; Molinski, 2002; Kappenman, 2007). In the worst cases, wide-spread blackouts and permanent damage of transformers may occur.
GIC in a power network can be investigated by measurements or by theoretical calculations. GIC values vary spatially much from site to site in a network. Therefore GIC data recorded at one place does not provide information about GIC values at other locations. Thus in practice, model calculations are needed to obtain a full overall understanding of GIC magnitudes and of the resulting risks and threats to a power grid. The calculations can be verified and adjusted by GIC measurements at one or a few sites.
A GIC calculation consists of two separate parts (e.g. Pirjola, 2002): (1) Determination of the horizontal geoelectric field at the Earth’s surface (called the “geophysical part”), and (2) Computation of GIC produced by the horizontal geoelectric field in the network (called the “engineering part”). The geophysical part, which is independent of the network considered, may be solved by using Maxwell’s equations and the boundary conditions of the electric and magnetic fields. The input consists of information or assumptions about the ground conductivity and about the currents flowing in the ionosphere and magnetosphere or about the geomagnetic variations at the Earth’s surface. Practical techniques to be applied to the geophysical part are discussed, for example, by Viljanen et al. (2004).
Besides the geoelectric field, the network topology and resistances must be known for solving the engineering part. Time variations of the geomagnetic and geoelectric fields and of GIC are very slow compared to the 50/60 Hz frequency used in power transmission. Therefore, a dc treatment is applicable to the engineering part. Lehtinen and Pirjola (1985) present a matrix method for determining the earthing GIC flowing between a power grid and the Earth as well as GIC in transmission lines. The basic equation expresses the earthing GIC as a column matrix in terms of another column matrix that consists of so-called “perfect-earthing” currents. They involve the dependence on the geovoltages accompanying the horizontal geoelectric field. To the author’s knowledge, the properties of the transfer matrix between the two column matrices have not been discussed in the GIC literature except for a brief study by Pirjola (2009). In this paper, we go deeper into this investigation with the aim to be able to draw physical conclusions from the eigenvalues and eigenvectors of the transfer matrix. The whole subject of matrices and their eigenvalues and eigenvectors refers mathematically to linear vector spaces. In this paper, however, discussions about abstract mathematical concepts and details are minimised.
The technique presented by Lehtinen and Pirjola (1985) is summarised in Section 2 including the definition of the transfer matrix in terms of network parameters. Eigenvalues and eigenvectors of the transfer matrix are discussed in Section 3. We consider the case of a three-node network in detail because it is simple enough to allow exact calculations and equations in practice. It is evident that conclusions can be drawn from the three-node case that might be extended and generalised to also hold true for more complex networks. Other numerical results presented in Section 3 concern the Finnish 400 kV network, which is introduced as a GIC test model by Pirjola (2009). In Section 3, special attention is paid to the norm of the earthing GIC column matrix, which provides a measure of GIC impacts on the system. Concluding remarks are given in Section 4.
2. Calculation of Geomagnetically Induced Currents in a Power Network
When GIC in a power network are calculated, it is convenient to treat the three phases as one circuit element. The resistance of the element is then one third of that of a single phase, and the GIC flowing in the element is three times the current in a single phase. Moreover, the earthing resistances (which might be called the total earthing resistances) are assumed to include the actual earthing resistances, the transformer resistances and the resistances of possible reactors or any resistors in the earthing leads of transformer neutrals.
It should be noted that knowing the currents J e,m and the resistances R n,im does generally not determine the voltages V im uniquely (i, m = 1, …, N) from Eq. (4). This can easily be seen, for example, by considering a simple three-node system in which all three line resistances are equal to Rn,12 = R n ,23 = Rn,31 = 1 Ω. The two different voltage sets (V12 = 0V, V23 = −200 V, V31 = 100 V) and (V12 = 100 V, V23 = −100 V, V31 = 200 V) give the same currents = 100 A, Je,2 = 200 A and Je,3 = −300 A.
3. Eigenvalues and Eigenvectors of the Transfer Matrix C = (1 + Y n Z e )−1
3.1 General situation
The norm of the pe currents (=∣J∣) is naturally defined with a similar formula. If J is an eigenvector of C with the eigenvalue λ, Eq. (11), together with the inequalities (12), shows that ∣I∣≤∣J∣, and normalising the column matrix J to unity, i.e. ∣J∣ = 1, ∣I∣ equals λ. It is worth noting that the unit of I and J is [A] whereas λ is dimensionless. Thus, to be precise, we should actually say that ∣J∣ = 1 A and ∣I∣ equals λ A. In this paper, however, no confusion can arise even if the use of the units is somewhat inaccurate.
3.2 Three-node network
3.2.1 Arbitrary resistances
Now we consider a three-node network (i.e. N = 3), which is simple enough to enable analytic calculations, but it does not lead to trivial conclusions as a two-node system would do. Let us number the nodes by 1, 2 and 3 and assume that their earthing resistances are S1, S2 and S3, respectively. The line resistance between the nodes 1 and 2 is denoted by R1, between the nodes 2 and 3 by R2 and between the nodes 3 and 1 by R3. (Thus, compared to Section 2, the notation is simplified by omitting the subscripts ‘e’ and ‘n’ and by using the symbol ‘S’ distinguishing the earthing resistances from the line resistances more clearly. The line resistances also have only one subscript now.) We assume that the nodes are sufficiently distant to permit the assumption that the earthing impedance matrix Z e is diagonal with S1, S2 and S3 being the diagonal elements. Pirjola (2008) indicates that this is generally not a serious or critical limitation in practical GIC studies.
The fact that no unique solution is obtained for all elements , but and depend on , which remains arbitrary, is in agreement with the statement about βJ being also an eigenvector in Section 3.1. Mathematically this is a consequence of Eq. (19). By using Eq. (27) and doing some algebraic work, it is possible to show that the sum is proportional to . Thus, based on Eq. (20), the sum is zero when λ k ≠ 1, which means that we have explicitly shown that Eq. (6) holds true for the eigenvectors J2 and J3 in an arbitrary three-node network.
3.2.2 Resonance case
Taking into account the definition of the S j and R j resistances (j = 1, 2, 3) in the beginning of Section 3.2.1, Eq. (28) mean that the products of an earthing resistance and the opposite line resistance are equal in the resonance case.
We see from Eq. (30) that any matrix is an eigenvector of C with the eigenvalue if J1 + J2 + J3 = 0, or in other words, all pe current 3 × 1 column matrices satisfying the physical equation (6) are eigenvectors of the transfer matrix C associated with the eigenvalue (Eqs. (29a) and (29d)). It is also possible to conclude from Eq. (30) with Eq. (29a) that a matrix J that satisfies the formulas and is an eigenvector corresponding to the eigenvalue λ1 = 1 (note the superscript “1” in the elements of the eigenvector). Using the definition of the α jk quantities and the resonance conditions (26), we obtain the relations and , which indicate that the pe currents connected with the eigenvalue λ1 = 1 are inversely proportional to the corresponding earthing resistances and proportional to the resistances of the opposite lines. These formulas further show, for example, that (when J ≠ 0), so that the physical condition (6) is not satisfied.
The physical contents of the resistance ratios in the last two expressions are that the earthing resistances of the nodes are divided by the resistance of the line next to this particular node. In the first and second form, we take the next line in the direction Node 1 → Node 2 → Node 3 → Node 1 and Node 1 → Node 3 → Node 2 → Node 1, respectively. Other expressions for λ2 and λ3 can also be derived, as for example λ2 = λ3 = (1 + gf)−1 where g = S1R2 = S2R3 = S3R1 (see Eq. (28)) and
A special situation of the resonance case is obtained if we assume that the earthing resistances are equal (S1 = S2 = S3) and the line resistances are equal (R1 = R2 = R3). Then all α jk quantities are the same (= α). By utilising the equations mentioned above in this section, we directly see, for example, that the eigenvalues are and that the elements of the eigenvector associated with λ1 are equal .
3.2.3 Numerical results
In the present case N equals 3, but Eq. (32) is naturally applicable to other values of N as well. Figure 1(b) shows that all possible column matrices J are really considered in the 32660 runs because the scalar product gets all values between −1 and +1. In other words, this means that J is rotated through the unit circle in the plane defined by the requirement that J1 + J2 + J3 = 0. (Referring to the comment about the units at the end of Section 3.1, we should actually say that J and Jref are normalised to 1 A making [A2] be the unit of the scalar product.)
The eigenvalues of C are λ1 = 1, λ2 = 0.1869 and λ3 = 0.2978, the two latter of which are indicated by the horizontal lines in Fig. 1(a). We see that the norm of the earthing currents mostly lies between λ2 and λ3 but not always. In Fig. 1(a), the minimum and maximum of ∣I∣ are 0.1867 and 0.2982, respectively. In practice, there is no difference between λ2 and min(∣I∣) and between λ3 and max(∣I∣). However, the positive differences λ2 − min(∣I∣) and max(∣I∣) − λ3 are larger than the possible numerical inaccuracy in the computations. In the calculations presented in Fig. 1(a), altogether 0.79% and 6.66% of the ∣I∣ values are below λ2 and above λ3, respectively. Thinking about practical GIC research, ∣I∣ gives a kind of an overall average impact of GIC on transformers, and the present calculation indicates that λ2 and λ3 determine the lower and upper limit of this impact. In this connection, it is necessary to emphasize that because J is normalised to unity the values of ∣I∣ are quite small in Fig. 1(a). In a practical GIC case, ∣J∣ is much larger than 1 A, and λ2∣J∣ and λ3∣J∣ give the limits for ∣I∣. This comment also concerns the numerical calculations presented below in this paper.
The eigenvalues λ2 = 0.2525524 and λ3 = 0.2919066 of C are shown by the horizontal lines in Fig. 2(a), and we see that the values of the norm ∣I∣ cover the area between λ2 and λ3 very well. However, if the values are considered exactly, we can again note that ∣I∣ can be slightly lower than λ2 or slightly larger than λ3 since min(∣I∣) and max(∣I∣) are 0.2525465 and 0.2919134, respectively. Similarly to Fig. 1(a), the positive differences λ2 − min(∣I∣) and max(∣I∣) − λ3 are, on one hand, insignificant in practice, but on the other hand, cannot be explained by numerical inaccuracies in the computations. Anyway, Fig. 2(a) supports the conclusion already drawn from Fig. 1(a) that λ2 and λ3 express the lower and upper limit of the practical GIC impact on the network.
Choosing the resistances to be S1 = 0.50 Ω, S2 = 7.50 Ω, S3 = 0.75 Ω, R1 = 0.80 Ω, R2 = 1.20 Ω and R3 = 0.08 Ω, we have the resonance case discussed in Section 3.2.2 since Eq. (28) are satisfied (with the products equalling 0.60 Ω2). According to the theory, the eigenvalues λ2 and λ3 of C are the same, and all column matrices for which J1 + J2 + J3 = 0 are corresponding eigenvectors. The value of λ2 = λ3 can be calculated from Eq. (31) to obtain 0.057971. The fact that λ2 = λ3 is much smaller than the values of λ2 and λ3 in the cases investigated in Figs. 1(a) and 2(a) obviously results from the large resistance S2. Decreasing S2 by a factor of 4 and increasing R3 by the same factor, i.e. S2 = 1.875 Ω, R3 = 0.32 Ω, so that Eq. (28) still hold true, give a clearly larger value λ2 = λ3 = 0.18079.
3.3 Finnish 400 kV test model
Pirjola (2009) introduces an old version of the Finnish 400 kV power network valid in 1978 to 1979 as a test model for GIC calculation algorithms and programs. The geographical structure of today’s 400 kV grid in Finland is quite similar to the particular old version though not the same. Regarding GIC, an essential difference is produced by the installation of series capacitors in north-south lines (Elovaara, 2007). They block the flow of the dc-like GIC, which means that the Finnish 400 kV network consists of two separate parts as concerns GIC nowadays. (It should be noted that the reasons for the use of series capacitors are other than GIC mitigation.)
The Finnish 400 kV test model has 17 stations and 19 transmission lines, thus being complex enough to enable realistic GIC calculations but not too large to unnecessarily hamper the computations. The Cartesian flat-Earth coordinates and (total) earthing resistances of the stations and the line resistances included in the test model are provided by Pirjola (2009). A special feature is that the earthing resistances of the two northwestern end stations are zero to account for the galvanic connection of the grid to the Swedish 400 kV network. It is additionally assumed that the stations are distant enough, so the earthing impedance matrix is diagonal. As Pirjola (2008) points out, it is not an essential limitation in practice. For reference information to the users of the test model, Pirjola (2009) also presents the values of GIC flowing to (or from) the Earth at the stations and of GIC in the transmission lines when the test model is impacted by a northward or an eastward uniform geoelectric field of 1 V/km.
An interesting detail concerning the test model is that the eigenvalue λ1 = 1 is a double value, so it has two independent eigenvectors. Looking at the issue more carefully, we see that the eigenvectors correspond to situations where the pe current has a non-zero value at one of the two northwestern zero-resistance stations and is zero elsewhere. Considering such a case in terms of external injection of the pe currents as mentioned in Section 2, the situation simply means that a current injected at a zero-resistance station directly flows into the Earth and nothing happens at other sites of the network. Thus, this is in accordance with what is mentioned about the pe currents associated with the eigenvalue equal to one in Section 3.1. It is easy to understand physically but does not have any practical interest or importance.
Besides Manitoba (Canada), similar calculations have also been performed for the high-voltage power networks in Brazil, China (ultra-high-voltage) and Southern Sweden. In every case, the inequalities (12) are valid as expected. Also otherwise, the results are comparable to those for the Finnish 400 kV test model. These additional results support the above-mentioned conclusion valid for the Manitoba case that an increase of the number of nodes makes the plots corresponding to Figs. 3(a) and 3(b) concentrate in smaller ranges when the calculation program is run (only) 100000 times. A detail worth noting as well is that, similarly to the three-node network and the Finnish 400 kV test model, the value λ = 1 (with a high accuracy) seems to be included in the set of eigenvalues in each calculation at least once. Finally, we emphasise that an earthing impedance matrix with non-zero off-diagonal elements (which is not the case for the three-node and test model cases) is included in the Manitoba and Southern Sweden computations. Because the conclusions are similar from all calculations, we thus obtain additional support to the fact that assuming the earthing impedance matrix to be diagonal is not a serious restriction in practice.
4. Concluding Remarks
Geomagnetically induced currents (GIC) flowing in power networks due to space weather can be calculated by using convenient matrix formulas whose input data consist of the horizontal geoelectric field at the Earth’s surface and of the network resistances and topology. This paper is focussed on the transfer matrix between the so-called “perfect-earthing” (pe) currents and GIC flowing between the Earth and the network. The pe currents depend on the geovoltages impacting the transmission lines and produced by the geoelectric field. The transfer matrix depends on the network resistances and topology in terms of an earthing impedance matrix and a network admittance matrix.
This paper provides a novel approach to the transfer matrix by considering its eigenvalues and eigenvectors. It is shown that either an eigenvalue equals one or the corresponding eigenvector satisfies the condition that the sum of the pe currents included in the particular eigenvector is zero, which implies that the amounts of pe currents flowing to and from the network are equal. The situation in which the eigenvalue equals one seems to be unimportant regarding practical GIC applications.
By physical arguments, we conclude in this paper that all eigenvalues of the transfer matrix are non-negative and less than or equal to one. This statement is proved also mathematically in the case of a three-node network, which is simple enough to enable analytic calculations related to eigenvalues and eigenvectors of the transfer matrix. It also seems evident that some conclusions drawn from the three-node case might be generalised to larger networks as well. In this paper, numerical computations that concern the Finnish 400 kV GIC test model also support that the eigenvalues lie between zero and one. The same eigenvalue range is indicated by numerical calculations for other power grids as well.
Special attention is paid to the norm (= square root of the sum of the squares) of the earthing GIC flowing between the Earth and the network. It is a quantity that gives an overall idea of the possibility of adverse impacts of GIC on a power grid since it is associated with GIC flowing through transformers. If the pe current matrix is adjusted to have the norm equal to one, the norm of the earthing GIC seems to have the lower and upper limits equal to the smallest and largest (but ≠1) eigenvalue of the transfer matrix. This is an observation valid in practical GIC studies but not exactly. Without the normalisation of the pe current matrix to one, the lower and upper limits equal the smallest and largest eigenvalue multiplied by the norm of the pe currents.
The author wishes to thank Dr. Ari Viljanen (Finnish Meteorological Institute) for many useful discussions on the topic of this paper. Thanks also go to all colleagues in the collaboration concerning the Finnish, Swedish, Manitoba, Brazilian and Chinese power networks discussed in this paper.
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