A NOVEL METHOD FOR THE CALCULATION OF SELF AND MUTUAL IMPEDANCES OF OVERHEAD CONDUCTORS AND PIPELINES BURIED IN TWO-LAYER SOILS.

Transcription

1 A NOVEL METHOD FOR THE CALCULATION OF SELF AND MUTUAL IMPEDANCES OF OVERHEAD CONDUCTORS AND PIPELINES BURIED IN TWO-LAYER SOILS. D. A. Tsiamitros, G. C. Christoforidis, G. K. Papagiannis, D. P. Labridis, P. S. Dokopoulos Aristotle University of Thessaloniki, Greece. ABSTRACT The influence of earth stratification on the impedances of overhead conductors and underground pipelines is examined in this paper. New formulas for the self and mutual earth return impedances for the two-layer earth case are derived. The involved semi-infinite integrals are calculated by a novel, numerically stable and efficient integration scheme. A typical transmission line-underground pipeline arrangement is examined for six cases of two-layer earth models over a wide frequency range. The accuracy of the results is justified by a proper Finite Element Method formulation. The results for the two-layer earth case are compared to the corresponding for the homogeneous earth case, showing significant differences. The results obtained by the new expressions, when implemented for the calculation of the homogeneous earth impedances are also compared to other approaches. Index Terms. Power System Analysis, Electromagnetic Transients, Power System Modelling and Simulation. INTRODUCTION The accurate calculation of the self and mutual impedances of overhead transmission lines and nearby buried metal pipelines is the first and most important step for the study of the electromagnetic interference between them. For the case of overhead lines, the conductor earth return impedances may be calculated using the widely accepted Carson s formulas []. Pollaczek [] has proposed similar formulas applicable to cases of underground conductors and to combinations of underground and overhead conductors. In these approaches earth is assumed to be semi-infinite and homogeneous. In practice, earth is composed of several layers of different electromagnetic properties. Sunde [3] and Nakagawa [4] proposed a solution for the case of overhead conductors above a multi-layered earth model. However, there is still lack of an analytic formulation for the case of overhead and underground conductors in multi-layered soil. This paper presents new analytic expressions for the calculation of the self and mutual earth return impedances of an overhead-underground conductor system in a two-layer earth. For the self and mutual impedances of the overhead conductors, the existing formulas given by Sunde [3] are adopted. However, the new expressions are suitable for the calculation of the mutual earth return impedance between each one of the overhead conductors and the underground pipe, as well as for the self earth return impedance of the pipe. The formulation was derived using a methodology based on Sunde s approach [3]. The new expressions have similar form to Pollaczek [] and Sunde [3] formulas, i.e. they also contain semi-infinite integral terms. The latter are calculated by a novel numerical integration scheme. This scheme is based on a combination of Gauss-Legendre and Gauss-Laguerre methods [5] with the Lobatto rule [6], in order to overcome efficiently the problems arising from the highly oscillative form of the infinite integrals. The results obtained by the new expressions are compared against those obtained by the application of the Finite Element Method (FEM for a system comprising of a typical single-circuit overhead transmission line and a pipeline buried in different twolayer earth structures based on actual ground resistivity measurements [7]. The FEM is a numerical method widely used for the solution of electromagnetic field equations in a region, regardless of geometric complexity. In a recently proposed method by some of the authors [8], [9] a suitable FEM formulation was used for the calculation of the self and mutual impedances between overhead transmission lines and underground metal pipelines. The method is capable of handling cases of terrain surface irregularities and nonhomogeneous, stratified soil, where most classical methods usually fail. The differences in the impedances between six twolayer earth models and the homogeneous earth case are presented. The homogeneous earth is assumed to have a resistivity equal to the first layer of the two-layer earth models. Finally, the new formulas are applied for the

2 calculation of the homogeneous earth return impedances and the results are compared to those obtained by the well-known ElectroMagnetic Transients Program (EMTP [0]. Significant differences are observed due to the approximations adopted in the latter reference. PROBLEM FORMULATION AND SOLUTION In Fig. an overhead conductor-underground insulated metal pipe configuration is shown. The depth of the pipe i is h while h is the overhead line distance from the surface of the earth. The horizontal separation distance between them is y ik. The depth of the first layer is d and the second layer is considered to be of infinite depth. The first layer has permeability µ, permittivity ε and conductivity σ, while the corresponding properties of the second earth layer are µ, ε, σ. The air has a conductivity σ 0 equal to zero and permeability µ 0 and permittivity ε 0 equal to those of the free space. Π 0X Π X Overhead line k z h Insulated pipe i y ik h Π X x y second earth layer µ ε σ Fig.. Geometric configuration of an overhead line and an underground insulated metal pipe in a two-layer earth. Field derived from a dipole located in the place of the overhead line. d air µ 0 ε 0 σ 0 =0 first earth layer µ ε σ According to Sunde [3], mutual impedances between wires may be derived by integrating the field due to dipoles. An assumption adopted in [3] is that the attenuation along the wires can be neglected at low frequencies. In this approach, the Π function is used for the derivation of the self and mutual impedance formulas. The Π function has been also adopted for the solution of the electromagnetic field in this paper. Assuming a horizontal dipole in the place of line k in Fig., the x-components of the Π function derived from that dipole must be found. The x-components of the Π function in the air, in the first and the second earth layer are defined as Π ox, Π x and Π x respectively and are given by [3]. Applying the boundary conditions at z=d and z=0 as given in [4], the field at any point can be fully defined. SELF AND MUTUAL IMPEDANCE FORMULAS A. Overhead Line Self And Mutual Impedance Formulas According to [3] and [4], the per unit length mutual impedance Z kj between the overhead line k and a conductor j located above the earth surface at z=d+h 3, y=y kj is found by integrating Π ox along the infinite line k, i.e. along the x-axis: ' Π ' ox( z= d + h3, y= ykj Zkj = γ 0 dx ( IdS In (, γ ο = jωµ 0( σ0 + jωε0 = ω µ 0ε0, ω=πf is the angular velocity, j is the imaginary unit and IdS is the moment of the dipole. The final form of the mutual impedance as given in [3] is shown in (. The self impedance of the overhead line k results from (, by replacing h 3 with h and y kj with zero In (, α = u + γ, γ = jωµ ( σ + jωε for the first layer and j ( j γ = ωµ σ + ωε for the second. α = u + γ, ' jωµ cos( uy 0 kj uh h u( h + h kj = + π 0 u 3 3 Z e e du y 0 kj ( h h3 jωµ + + ln( j = + π ykj + ( h h3 ( µ + µα + ( µ µα ad jωµ 0µ cos( uy kj a a e uh ( + h3 e du= π ad 0 ( µ a + µα ( µ 0 a + µ u + ( µ a µα ( µ u µ 0 a e Fu ( = ωµ 0µ uh ( + h3 Fue ( cos( uykj du π ad ( µ a+ µα + ( µ a µα e ( µ + µα ( µ + µ + ( µ µα ( µ µ a a u a u a e ad ( (3

3 3 B. Pipe And Overhead Line Mutual Impedance Formula The per unit length mutual impedance Z ik between the pipeline i and the overhead line k is found by integrating Π x along the infinite line k, i.e. along the x- axis: ' Π ' ( x z= d h, y= yik = γ dx (4 IdS The final form of the mutual impedance is shown in (5. If it is assumed that the first and second earth layer electromagnetic properties are equal, (5 becomes: ah uh ' jωµ µ 0 e e = { cos( uyik du π 0 ( µ 0α + µ u (6 Equation (6 is the same proposed by Pollaczek [] for the calculation of the mutual impedance between an underground conductor and a conductor over a homogeneous earth. Finally, if the first earth layer has identical electromagnetic properties with the air, Carson s formula for the mutual impedance between overhead lines over homogeneous earth is derived from (5. C. Pipeline Self Impedance Formula In order to determine the pipe self impedance, the x- components of the Π function derived by a dipole in the place of the pipe must be defined. This can be done using the reciprocity theorem []. This theorem implies that in a linear, bilateral, single circuit network, the ratio of excitation to response is constant, when the position of excitation and response are interchanged. However, the ratio of the voltage induced on the conductor i to the current imposed on the conductor k is shown in (5. This ratio should be equal to the ratio of the voltage induced on the conductor k to the current imposed on the conductor i. From this equalization, the x-components of the Π function derived by the pipeline dipole can be obtained. Following the above reasoning, the overheadunderground conductor configuration of Fig. is considered, where r ii is the outer radius of the pipe. The Π function x-components of the pipeline dipole in the air, in the first and the second earth layer are defined as Π ox, Π x and Π x respectively. The per unit length mutual impedance Z ik between the pipe i and the overhead conductor k is also given by: Π ' ox( z= d + h, y= yik = γ o dx IdS Π 0X Π X Π X Overhead line k h y ik r ii x z h Insulated pipe i y d air µ o ε o σ o =0 (7 first earth layer µ ε σ second earth layer µ ε σ Fig.. Geometric configuration of an overhead line and an underground insulated metal pipe in a two-layer earth. Field derived from a dipole located in the place of the pipe. From (5 and (7, Π οx is completely defined. In order to define Π x, a similar procedure is adopted. Assuming that a cable m is located in the second earth layer instead of the air, Π x can be found. Finally, using the boundary conditions given in [4] at z=d and z=0 in Fig., the field Π x in the first earth layer is found. Thus, the per unit length self impedance Z ii of the pipe is calculated by: Zii Π ( x z= d h, y= r ii = γ dx IdS The final form of (8 is given in (9. (8 ah uh a( d h uh ' jωµ µ 0 ( µα + µα e e + ( µα µα e e = cos( uy ad ik du π 0 ( µα 0 + µ u( µα + µα + ( µα µα ( µ u µα 0 e (5 Z ii 0 s s + s d e ( + d s e + d d e ad s0s d0de cos( 0 0 a d h ii 0 ah 0 ad jωµ ur = π α du (9

4 4 In (5 and (9, αο = + γ, s = ( µ α + µ α, u = ( +, d ( µ α µ α = ( µ α µ α. s µ α µ α d0 0 ο ο = and Further observation of (5 reveals that it essentially consists of two terms: The first and dominant one contains two exponential functions which represent the influence of the vertical distance between the underground pipeline and the overhead conductor. These exponential functions have as a factor the sum of the electromagnetic properties of the two earth layers. The second term depends on the distance between the overhead conductor and the image of the pipeline with respect to the two earth layers limit. The factor of this term is the difference between the electromagnetic properties that determine the two earth layers. Thus, when the resistivity, permeability and permittivity values of the two earth layers are equal, this term vanishes, as the boundary of the two layers disappears too. On the other hand, further observation of (9 reveals that it essentially consists of four terms: The first and dominant one contains two terms denoted with the symbol s to indicate the sum of electromagnetic properties of three different areas: the air, the first and the second earth layer. The second term depends on the distance between the pipe and its image with respect to the two earth layers limit. One factor of this term is represented with the symbol d to indicate the difference between the electromagnetic properties that determine the two earth layers. Thus, when the resistivity, permeability and permittivity values of the two earth layers are equal, this term vanishes, as the boundary of the two layers disappears too. The third term refers to the distance between the pipe and its image with respect to the air-earth boundary. One of the factors of the latter term is d 0, indicating the difference between the electromagnetic properties that determine the air-ground boundary. When this boundary is absent, this term is also zero. Finally, the fourth term contains an exponential function which depends on the distance between the two images of the pipe with respect to the two boundaries of Fig.. As expected, this term has as factors only symbols denoted with d, because it vanishes whenever one of the two boundaries does not exist. NUMERICAL INTEGRATION OF THE IMPEDANCE FORMULAS Direct numerical integration is used for the calculation of the semi-infinite integrals in (, (5 and (9. The integrals are highly oscillatory, showing also an initial steep descent. For these reasons, the use of a single numerical integration method proved to be inefficient. To overcome these difficulties, a combination of numerical integration methods was implemented. More specifically, the Gauss-Legendre method [5], a highly accurate numerical integration method applicable in finite intervals of functions, is combined with two other methods: the Gauss-Laguerre method [5], which is best suited for infinite integrals and the Lobatto rule, a very efficient method for oscillative functions [6]. The selective implementation of the different integration methods in the intervals between the roots of the cosine function of the integrals leads to a quick and very efficient integration scheme for the evaluation of (, (5 and (9. This novel integration scheme was tested in the numerical calculation of highly oscillative infinite integral terms in [] and was applied successfully to the calculation of the earth return impedances in cases of homogeneous earth [3]. The new integration method showed remarkable numerical stability and efficiency, in all cases examined. THE FINITE ELEMENT APPROACH The FEM package [4], developed at the Power Systems Laboratory of the Aristotle University of Thessaloniki during the last 6 years, has been used for the finite element formulation of the cases under investigation. A local error estimator, based on the discontinuity of the instantaneous tangential components of the magnetic field, has been chosen for an iteratively adaptive mesh generation, as in [4]. NUMERICAL RESULTS A. Comparison With The FEM Six different two-layer earth models, based on actual grounding parameter measurements [7], are investigated. The corresponding data for the resistivities ρ of the first and ρ of the second layer and for the depth d of the first layer are shown in Table I. The second layer is considered to be of infinite depth. A typical single-circuit overhead transmission line with

5 5 two ground wires is examined. The line configuration is shown in Fig. 3 as well as the pipe s position. The ground and the phase wires are considered to be solid cylindrical conductors with radius and m respectively. The ground wires have conductivity equal to 3.5*0 +6 S/m and relative permeability 50. The phase wires corresponding values are 3.65*0 +7 S/m and. The insulated metal pipe conductor has inner radius equal to 0.95 m and outer radius 0. m, while the insulation thickness is 0. m. The conductivity and the relative permeability of the pipe are the same with those of the ground wires. The insulation of the pipe has relative permeability equal to, while the relative permeability and permittivity of the air are those of the free space. 3 m is defined at 0-7, the computational time is less than 8 min for the set of the 60 cases. The results are compared to the corresponding ones obtained by the FEM [4] for further justification. The relative differences using (0 are calculated as: Z formula ZFEM Relative difference (% = 00 (0 ZFEM In Fig. 4 the relative differences for the magnitude of the mutual impedance between the pipe and the closest phase wire of the overhead line are shown. 5% 4% CASE I CASE III CASE II CASE IV 6 m 6 m 6 m m 4 m air µ 0 ε 0 σ 0 =0 difference (% 3% % % CASE V CASE VI Insulated pipe i 5 m. m d first earth layer µ ε σ,e+00,e+0,e+0,e+03,e+04,e+05,e+06 frequency (Hz Fig. 4. Differences in the magnitude of the mutual impedance between the formula and the FEM. second earth layer µ ε σ Fig. 3. Geometric configuration of a single circuit overhead line with two ground wires and an underground insulated metal pipe in a twolayer earth. TABLE I TWO-LAYER EARTH MODELS. ρ (Ωm ρ (Ωm d(m CASE I CASE II CASE III CASE IV CASE V CASE VI Series impedances are calculated for the configuration of Fig. 3 and the six earth models of Table I using the proposed formulas of (, (5 and (9 for the frequency range of 50 Hz to MHz. The numerical integration techniques used for the calculation of the proposed formulas proved to be numerically stable. The computation time for the numerical integration, when the defined tolerance is set equal to 0-8, is less than 5 min for the derivation of the impedance matrices of a set of 60 resistivity and frequency combinations using an Intel Pentium IV PC at.66ghz. When the tolerance In all cases examined the differences recorded are less than % for the whole frequency range under consideration. B. Comparison With The Homogeneous Earth Results The series impedances calculated for the configuration of Fig. 3 and the six earth models of Table I using the proposed formulas are compared to those obtained when earth is considered to be homogeneous for the frequency range of 50 Hz to MHz. The differences calculated by ( are shown in Fig. 5 and Fig. 6 for the magnitude of the self impedance of the pipe and the mutual impedance between the pipe and the closest phase wire respectively. Differences almost up to 75 % appear. ( Ztwo layer Zhom ogeneous Rel. difference (% = 00 ( Z hom ogeneous In (, Z homogeneous is the magnitude of the series impedance obtained for the configuration of Fig. 3 when all the formulas proposed are simplified for the homogeneous earth case. The resistivity of the homogeneous earth is considered equal to the resistivity ρ of the first earth layer.

6 6 Fig. 5 and Fig. 6 indicate that the differences between the homogeneous and the two-layer earth models increase in cases of great divergence between the resistivities of the two layers. difference (% CASE I CASE III CASE V CASE II CASE IV CASE VI -,E+00,E+0,E+0,E+03,E+04,E+05,E+06 frequency (Hz range of 50 Hz to MHz and for earth resistivities from 0 up to 000 Ωm. The results are compared to those obtained by the EMTP CABLE CONSTANTS Supporting Routine. The relative differences calculated by a formula similar to (0 for the magnitude of the mutual impedance between the pipe and the closest phase wire of the overhead line are presented in Fig. 7. difference (% ρ=0 Ωm ρ=50 Ωm ρ=00 Ωm ρ=300 Ωm ρ=500 Ωm ρ=000 Ωm Fig. 5. Differences in the self impedance of the pipe between the twolayered and the homogeneous earth case.,e+00,e+0,e+0,e+03,e+04,e+05,e+06 frequency (Hz 9 CASE I CASE III CASE II CASE IV Fig. 7. Differences in the mutual impedance of the pipe and the overhead line between EMTP and the new numerical integration method. difference (% 6 3 CASE V CASE VI As expected, the differences are maximized for high frequencies and low earth resistivities, reaching up to 80 %. -3 CONCLUSIONS -6,E+00,E+0,E+0,E+03,E+04,E+05,E+06 frequency (Hz Fig. 6. Differences in the mutual impedance between the pipe and the overhead line between the two-layered and the homogeneous earth case. C. Comparison With The ElectroMagnetic Transients Program Equation (6 is the equation proposed by Pollaczek [] for the mutual impedance between a conductor buried in a homogeneous earth and an overhead line. In the wellknown Electromagnetic Transients Program (EMTP [0], the infinite series approximation proposed by Carson [] is used for its calculation, after the assumption that α = u + γ u. This of course is a highly uncertain assumption especially for high frequencies and low earth resistivities and may lead to significant errors as reported in [3]. Using the new numerical integration method adopted in this contribution, (6 is calculated for the frequency The problem of the calculation of the earth return impedances of systems composed of overhead lines and underground metal pipelines buried in a two-layered earth is addressed in this paper. Analytic expressions for the self and mutual impedances are derived. Replacing the parameters with those corresponding to the semiinfinite homogeneous earth structure, the expressions are transformed to the well-known classical Carson and Pollaczek formulas. For the numerical evaluation of the semi-infinite integrals involved in the derived expressions, novel numerical integration schemes, based on proper combinations of integration methods, are implemented. The proposed formulation is applied for the case of a typical single-circuit overhead transmission lineunderground pipeline system, and a variable two-layer earth structure. Six different cases of two-layer earth configurations with different electromagnetic properties, based on actual ground resistivity measurements, are examined. Results are compared to those obtained by a FEM formulation. The differences in the self and mutual impedance magnitudes and arguments are less than % over a wide frequency

7 7 range. The results obtained by the new formulas show differences up to 75% against those for the homogeneous earth case when the resistivity of the earth is considered to be equal to the resistivity of the first layer. The results obtained by the new expressions for the homogeneous earth case, are compared to those obtained by the ElectroMagnetic Transients Program (EMTP. Significant differences up to 80 % occur, especially for high frequencies and low earth resistivities due to the approximations incorporated in the EMTP CABLE CONSTANTS supporting routine.. Papagiannis G. K., Tsiamitros D. A., Labridis D. P., Dokopoulos P. S., 00, MedPower00 Conference, Athens, Greece. 3. Papagiannis G. K., Tsiamitros D. A., Andreou G. T., Labridis D. P., Dokopoulos P. S., 003, IEEE Bologna Power Tech Conference Proceedings, 3, Labridis D. P., 000, IEEE Trans. On Magnetics, 36, Nr., REFERENCES. Carson J. R., 96, Bell Syst. Tech. J., 5, Pollaczek F., 96, Elektrische Nachrichtentechnik, 3, Sunde E. D., 968, Earth Conduction effects in transmission systems, nd ed., Dover Publications, 30-33, Nakagawa M., Ametani A., Iwamoto K., 973, Proc. IEE, 0, Nr., Ralston A., Rabinowitch P., 988, A First Course in Numerical Analysis, nd ed., McGraw-Hill, 00, Davies P., Rabinowitch P., 984, Methods of Numerical Integration, nd ed., Academic Press, 04, Alamo J. L., 993, IEEE Trans. On Power Delivery, 8, Nr. 4, Christoforidis G. C., Labridis D. P., Dokopoulos P. S., 003, Electric Power Systems Research, 66, Issue, Christoforidis G. C., Labridis D. P., Dokopoulos P. S., Kioupis N., 004, CEOCOR Meeting, Dresden, Germany. 0. Dommel H. W., 986, Electromagnetic Transients Program Reference Manual, Bonneville Power Administration, Portland, OR.. Dorf R. C., 993, The Electrical Engineering Handbook, CRC Press,