Influence of Earth Conductivity and Permittivity Frequency Dependence in Electromagnetic Transient Phenomena

Transcription

1 Influence of Earth Conductivity and Permittivity Frequency Deendence in Electromagnetic Transient Phenomena C. M. Portela M. C. Tavares J. Pissolato Deartamento de Engenharia Elétrica Faculdade de Engenharia Elétrica e de Comutação COPPE Universidade Federal do Rio de Janeiro Universidade de São Paulo UNICAMP Universidade Estadual de Caminas Rua Eng. Cesar Grillo, PO Box : 359 Zi : PO Box 6 Zi:38-97 Rio de Janeiro, RJ São Carlos, SP Caminas, SP BRAZIL Abstract In this article the quasi-modes model is used to observe the influence, in Electromagnetic Transient Phenomena, of considering a more accurate reresentation of soil, taking into account the earth conductivity and ermittivity frequency deendence. For an actual 44 kv three-hase transmission line the soil behavior is reresented through an unique real value of conductance and through a more accurate model, considering its electromagnetic behavior. Keywords: Soil model, Line arameters, Frequency deendence. I. INTRODUCTION The soil model resented satisfies the hysical coherence conditions in what concerns the relations between the conductivity (σ) and the ermittivity (ε) in the frequency domain. Some examles of ground arameters are resented. The effect of the soil behavior in some transmission line transients, and its influence in the overvoltages obtained, are shown. For an actual 44 kv three-hase transmission line the soil behavior is reresented through an unique real value of conductance and through a more accurate model of its electromagnetic behavior, considering the earth conductivity and ermittivity frequency deendence. In some cases, a roer earth model can lead to very different results than the ones obtained with simle real conductivity value, as shown. The influence of the frequency deendence of the soil arameters, in some line transient henomena, is resented. In the cases where the transients have high homoolar comonent, the effect of using a more accurate soil model can be very imortant. II. SOIL ELECTROMAGNETIC BEHAVIOR One essential asect of grounding systems study and simulation is the adequate soil modeling. Excet for very high electric fields, which originate significative soil ionization, soil electromagnetic behavior is essentially linear, but with electric conductivity, s, and electric ermittivity, e, strongly frequency deendent. The magnetic ermeability, m, is, in general, almost equal to vacuum magnetic ermeability, m. For slow variation of electromagnetic entities, it may occur an hysteresis tye behavior. For direct current or very slow variations of electromagnetic entities, it may occur humidity migration henomena, including electroosmosis and effects of temerature heterogeneity, which can not be dealt with only by means of local soil arameters. For fast transients, namely those associated to lightning, the soil behavior is imortant in a reasonably wide frequency range, tyically [, 2 MHz ]. Field measurements have some difficulties. After several trials with quite different soils, field and laboratory systematic measurements, namely varying water content of samles, we have develoed a measurement rocedure, considering the methods to collect soil samles, to measure arameters in function of frequency, and to obtain a hysically consistent model of soil electromagnetic behavior. As an examle of samle collection and measurement asects to coe with: it is necessary to assure the maintenance of soil structure and humidity, in samles, and to minimize effects of local soil heterogeneity. Three main rocedures have been develoed, for comact soils (namely those including clay), for ulverulent soils (including sand), and for rock. Basic descrition of such rocedures is resented in [-5]. Such rocedures have been alied, with good results, to a large number of sites and soil tyes, including remote sites. The field measurements of real soil have inherent disersion. A urely mathematical fitting may lead to hysical inconsistent models, with quite wrong results obtained with such models, e.g. by Fourier methods. It is adequate to have a robust validation criteria of soil models, covering real soil characteristics. In [-5] several soil electric models have been resented and justified, which : - Cover a large number of soil measured arameters, with good accuracy, and within the range of confidence of ractical field measurement. - Satisfy coherence conditions. In this aer the electric soil arameters are alied (s, w e ), in function of frequency, considering a articular set of the models described in [-5]. The arameters of such model were chosen according a minimum difference criteria, for field measured electric arameters, in function of frequency, for 68 ground samles in eight Sites, in Brazil, covering very different soil tyes and geological structures. The agreement of obtained models with measured arameters is within or near the confidence range of field measurement values. The measurements

2 were done in a frequency range [ Hz, 2 MHz]. In each Site, the maximum distance among ground oints at which samles where collected is less than 5 m. II.. Soil Models The model which have been used in resented results are some of models described in [-5]. With the excetion indicated below, the models, whose results are resented, are a sum of minimum hase shift arcels, W j, which aly to the immittance tye magnitude (in comlex or tensorial formulation of alternating magnitudes) W = s + i w e (w = 2 π f, being f the frequency) ( ) being i = + and m W = W ( 2) j j= All submodels used for W j are articular conditions of Tye 3 model described in [], resented below. Aart from slow henomena and hysteresis tye henomena, soil behavior is, tyically, of minimum hase shift tye. For a great number of soils, on frequency range [, 2 MHz ], in a first aroach, it is σ = a + b ω and ωε = c ω ( 3) being a, b, c, a constant arameters (frequency indeendent). For some soils, a similar behavior occurs, but for a smaller frequency range, e.g. [, khz ], and, for higher frequency, the behavior is different, namely with a lower ωe increase, or till a ωe decrease, when frequency increases. In order to analyze the frequency behavior of s, e, it is convenient to consider comlex formulation of electromagnetic entities, and to consider s + i ωe as an immittance. In fact, aart geometric factors, s + i ωe may be associated to the admittance of a volume element δv. Tye 3 model can be described as resented below, for which : W j (ω) = k. b ib a ia ( 4) 2F,, +, ω 2F,, +, ω reresenting 2 F [,,, ] the hyergeometric function, with four arguments, 2 F, according notation of [6]. This submodel has four indeendent arameters (k,, a, b). Considering, in this model (4), a =, the model becomes : W j (ω) = k b ω b,, +, ω i 2 F ib = k 2 F,, +, ( 5) ω Considering, in the model (4), a =, b and j =, the model becomes : π j W j (ω) = K j + i tang j ω ( 6) 2 A arcel W j as indicated in (6) is equivalent to arcel b ω of σ and to ωε = c ω, as indicated in () and (3), doing b = K j, π c = K tang and j 2 j j =, with the c π condition = tang. This condition has been verified b 2 in soil measurements, within measurement accuracy and soil heterogeneity effects. Considering, is this model (6), j =, the model becomes : s constant, ω e null ( ure conductor) ( 7) Considering, is the model (6), j, the model becomes : s null, w e roortional to w, e constant ( ure dielectric) ( 8) Within the range [, 2 MHz ], for all soil samles modeled in this aer, it is accurate enough to consider two arcels, for s + i w e, one constant (in most cases real), and another of tye (4) or of tye (5), frequency deendent. In a few cases, there is a net hysteresis effect, that can be modeled with an imaginary art of constant arcel. For all samles, a is the dominant arameter of the relative shae of frequency deendent arcel, W j, of s + i w e. For a = such arcel corresonds to a ure conductor ( s frequency indeendent, e null ). For a =, such arcel corresonds to a ure dielectric ( s null, e constant ). In all samles, for frequency deendent arcel, it is < a <. II.2. Statistical Distribution of Soil Parameters In order to allow a direct interretation of statistical distribution of main electric arameters of ground, in a way indeendent of model details, the following arameters were chosen, according the models adoted, indeendently, for the 68 soil samles, satisfying hysical coherence conditions: s = s ( Hz) D r = Ds = s ( MHz) - s ( Hz) s at Hz. s increase between Hz and MHz. D i = D(w e) = w e ( MHz) - w e ( Hz) w e increase between Hz and MHz. a arameter of frequency deendent arcel of s + i w e. It was verified that, for these samles, the two arcels of s + i w e, one constant, the other frequency deendent, are statistically indeendent. This fact, and the fact that no significative correlation exists between the air [D i, a ], but it exists between the air [ D r, a ], arises the hyothesis that: - The constant and frequency deendent arcels of s + i w e are related to quite distinct asects of hysical ground behavior. - The frequency deendent arcel is mainly associated to a dielectric hysical rocess, with related dissiative effects. Such dissiative effects are quite different of conductive behavior associated to constant arcel. In Figure we reresent the robability density,, of arameters s, D r, D i, a, considered searately, and, in Figure 2, the robability density,, of arameters [D i, a],

3 considered together, with Weibull aroximations based in the 68 soil samles. s [(ms/m) - ] r [(ms/m) - ] =.763 Soil 2 : s = A ; ω e = which results in ρ : 882 Ω.m, constant. The conductivity of the studied soils were chosen to be equal at low frequency, in order to comare the obtained results, taking into account that traditional measurement of soil resistivity is done at low frequency. Soil considers two arcels W j as described in II.3, namely one constant arcel, A,of the tye (7) and a frequency deendent arcel, B ω, of the tye (6). In soil 2, only the first arcel, A, is considered. III. CALCULATING THE LINE PARAMETERS In order to imlement the soil model, the line arameters were calculated using the aroximated formula which include the earth effect in longitudinal imedance as equivalent to have an ideal ground surface at a deth D (comlex) below hysical ground surface [7]. k d km i [(ms/m) - ] a Figure - Probability density,, of arameters σ, r, i,, considered searately, with Weibull aroximations based in the 68 soil samles. Scales of alicable to σ, r, i are graduated in (ms/m) -. h' k = h k + D' h k D km m h m D' = real ground ( σ+ i ωε) i ωµ a ideal ground h' m h' k i [(ms/m) - ] Figure 2- Probability density,, of arameters [ D i, a ], considered together, with Weibull aroximations based in the 68 soil samles and without correlation between D i and a. Values of, in white, are exressed in (ms/m) -. II.3. Soil Parameters Alied The soil arameters used in these examles were obtained from the exeriments described in [3], and are resented below : Soil : s + i ω e = A + B ω being A, B, constants, and A = 84.6 µs/m B = [ i] (µs/m) s Figure 3 - Conductors k and m osition suosing the earth at a comlex deth D. The transmission line longitudinal arameters are formed by : Z = Z km km k, m =, 2,..., n ( 9) where : Z km longitudinal imedance matrix element, er unit length; n total number of conductors and Z = Z int + Z ext ( ) where Z int conductor internal imedance, er unit length; Z ext conductor external imedance, er unit length; and Z where ωµ D km ext = i ln k, m =, 2,..., n ( ) 2π d km

4 D km and d km are defined schematically in Fig. 3, being : D' = ( 2) σ+ i ω ε i ω µ ( ) For the self terms (k = m) D km = 2h k ( 3) d km = r k ( 4) and Z int = R int + i X int ( 5) where R int internal conductor resistance X int internal conductor reactance In Figs. 4-5 the er unit longitudinal arameters for the transosed line using both soil models are resented. Resistance [ohm/km], Single three-hase transosed line soil - A µs/m B - [ i] (µs/m) s soil 2 - σ = 84.6 µs/m; ε = = β - soil zero - soil = β - soil 2 zero - soil 2, Frequency [Hz] Figure 4 Resistance er unit length comaring both soil models. Single three-hase transosed line soil - A µs/m B - [ i] (µs/m) s soil 2 - σ = 84.6 µs/m; ε = modes. Tyical cases in which frequency range above khz is imortant are : transients originated by lightning; front of wave asects of transients associated with shortcircuits, which may be quite imortant in what concerns insulation coordination. Table R and L er unit length values transosed line soil. Freq [Hz] R (Ω/km) L (mh/km) R h(ω/km) L h(mh/km) Table 2 R and L er unit length values transosed line soil 2. Freq [Hz] R (Ω/km) L (mh/km) R h(ω/km) L h(mh/km) IV. SINGLE THREE-PHASE LINE APPLICATION In Fig. 6 the data of the three-hase line used to illustrate the model are resented. The line arameters were calculated in the range of Hz to khz. As it is a single line, to reresent its modes (exact ones for transosed line and quasi-modes for non-transosed line) it was alied Clarke s transformation matrix, as exlained in [8-]. With the longitudinal imedance and transversal admittance in mode domain, the synthetic circuits were calculated, comosing one cascade of π-circuits for each mode, each reresenting km length. (7.5, 36.) 3.6 m Inductance [mh/km] = β - soil zero - soil = β - soil 2 zero - soil 2 (9.27, 24.7).4 m.4 m hase conductors : grosbeak ground wires : EHS - 3/8" line length : 4 km sag at midsan hase cond. : 3.43 m ground wires : 6.4 m Frequency [Hz] Figure 5 Inductance er unit length comaring both soil models. In Table and 2 some values of resistance and inductance er unit length are resented, for the transosed line. The difference between line arameters for the two soil models is imortant, namely for homoolar mode (e.g., 37 % in the longitudinal resistance er unit length, at khz). For fast transients, for which imortant frequency range may include frequencies above khz, the difference between the two soil models may be imortant also for non-homoolar modes. E. g., for khz, there is an order of magnitude difference in resistance er unit length, between the two soil models, for non-homoolar Figure 6 - Schematic reresentation of the 44 kv threehase line. The line was suosed transosed. Some transient studies were erformed in order to analyze the models, as resented. IV.. Frequency Analysis A frequency scan analysis was erformed for both models where the sending terminal had a V source and the receiving end was oened. The relations between the line ends were analyzed in the range of Hz to khz.

5 In Fig. 7 the zero sequence resonse is resented for the transosed line. The results for both soil models are discussed below : - The ositive sequence resonse was similar for both models. - The zero sequence resonse for the frequency deendant soil model is more damed than the one which uses only constant conductance. In Table 3 and 4 some values of zero sequence resonse are resented, for the transosed line. The difference between the zero sequence resonse for the two soil models is imortant, e.g. 4 % at Hz. Recetion voltage (module) [V] Frequency scan Transosed line soil 2 (mod) soil (mod) Frequency [Hz] Figure 7 - Zero sequence - Transosed line. Table 3 Zero sequence resonse transosed line soil. Freq [Hz] Module (V) Phase ( ) Table 4 Zero sequence resonse transosed line soil 2. IV.2. Mode Analysis Freq [Hz] Module (V) Phase ( ) The following test erformed with both models was to verify the natural mode behavior for a single three-hase transmission line, suosing it ideally transosed and nontransosed. The simulation consisted of alying a V ste of ms to verify the model behavior to transients in the frequency range of the normal switching henomenon. In Fig. 8 the diagram of the studied system is shown. Figure 8 - Simulated system for mode analysis. To reresent the modes the ste voltages were inut as described in Tab. 5. The recetion end was oened. In Fig. 9 some results of the mode analysis are resented. Analyzing the results it can be seen that : - Modes alha and beta had very similar results for both models, as could be seen from the revious results; - The homoolar mode resents some differences, which can imly in different overvoltages if the henomenon has high contribution of this mode. (a) (b) Recetion voltage [V] Recetion voltage [V] 2,,5,,5, -,5 -, -,5,5,,5, -,5 -, -,5 Table 5 - Stes to reresent the modes. Mode Phase Voltage (V) alha a -.5 b (central) +. c -.5 beta a +. b. c -. zero a. b. c. Homoolar mode Transosed line time [ms] Homoolar mode Transosed line soil 2 soil -2, Time [ms] soil 2 soil Figure 9 - Ste resonse for mode homoolar - Transosed line - (a) comlete simulation; (b) 2 ms simulation detail. V. CONCLUSIONS We have resented some basic asects of soil modeling, and shown that : - It is essential to choose a soil model that satisfies the hysical coherence conditions in what concerns the relations between the conductivity (σ) and the ermittivity (ε) in the frequency domain. - The soil behavior is, tyically, of minimum hase shift tye. - The usual assumtions about ground electric ermittivity, e. g. of the order of to 3 times vacuum electric ermittivity, and frequency

6 indeendent, is too far from reality, for most soils. - Soil arameters σ and ε are strongly frequency deendent. For most electrical engineering alications, s + i w e may be considered the sum of two statistically indeendent arcels, one constant, and the other frequency deendent. Such two arcels are associated to distinct hysical asects. - The frequency deendent arcel has a statistical disersion, among different soil tyes and conditions, much lower than the constant arcel. - The frequency deendent arcel may be defined by two arameters, which may be considered statistically indeendent. - There are imortant differences, sometimes of order of magnitude, among the induced voltages, electric field in ground, transferred voltages, according to the soil models, till among models with similar behavior at low frequency, and due to asects usually not considered, as it is the case of soil arameters deendence with frequency. So, it is essential, for most alications concerning grounding systems, or involving electromagnetic henomena affected by ground, to model adequately the ground behavior, including several asects not considered in common ractice. For transmission lines, according to secific conditions, and the henomena being studied, it may be quite imortant to model correctly the soil, considering frequency deendence of s + i w e. We have resented some illustrative results for a 44 kv three-hase transmission line. The soil behavior is reresented through two alternative soil models. In the first soil model we have considered an accurate soil reresentation, satisfying coherence conditions and with s + i w e frequency deendent. In the second soil model, we have considered a constant, frequency indeendent, conductance and ω ε much lower than σ. In some cases, an adequate earth model can lead to results quite different from those obtained with the usual rocedure of considering the arameter σ of soil frequency indeendent and arameter ε frequency indeendent with a relatively small value. The conditions in which such difference can be imortant include the following examles : - Switching conditions in which an imortant homoolar comonent may occur, either due to sread of switching of the three oles, or to fault conditions, and in which the imortant frequency sectrum is not restricted to extremely low frequencies ( < khz), and includes frequencies till about khz. - Network sustained oeration, faults and maneuvers in which it occurs conditions near resonance, for the homoolar comonent, for frequencies not restricted to extremely low frequencies ( < khz), and, e. g., for frequency between and khz. - Fast transients, for which imortant frequency range may include frequencies above khz. In this case, the difference between an accurate soil model and usual assumtions may be imortant also for nonhomoolar modes. Tyical cases in which frequency range above khz is imortant are: transients originated by lightning; front of wave asects of transients associated with short-circuits, which may be quite imortant in what concerns insulation coordination. VI. ACKNOWLEDGEMENT The measurement and modeling of soil samles of which artial results have been used in this aer, where done under a Contract of CISCEA, Comissão de Imlantação do Sistema de Controle do Esaço Aéreo (Air Sace Control System Establishment Commission) with Fundação COPPETEC. The authors thank CISCEA the ermission to use such results. The authors thank the suort received from FAPESP - Fundação de Amaro à Pesquisa do Estado de São Paulo. VII. REFERENCES [] Portela, C. Grounding Requirement to Assure Peole and Equiment Safety Against Lightning - Proceedings IEEE 2 International Symosium on Electromagnetic Comatibility, , August 2, Washington DC, United States. [2] Portela, C., Frequency and Transient Behavior of Grounding Systems - Physical and Methodological Asects, Proceedings 997 International Symosium on Electromagnetic Comatibility, , August 997, United States. [3] Portela, C., Frequency and Transient Behavior of Grounding Systems - II Practical Alication Examles, Proceedings 997 International Symosium on Electromagnetic Comatibility, , August 997, United States. [4] Portela, C., Statistical Asects of Soil Electromagnetic Behavior in Frequency Domain, Ground 2 - International Conference on Grounding and Earthing, Proceedings,. 99-4, June 2, Belo Horizonte, Brazil. [5] Portela, C. - Measurement and Modeling of Soil Electromagnetic Behavior - Proceedings IEEE 999 International Symosium on Electromagnetic Comatibility, IEEE Institute of Electrical and Electronic Engineers - EMC Society,. 4-9, August 999, United States. [6] Abramowitz, M., Stegun I., Handbook of Mathematical Functions, National Bureau of Standards, 964. [7] Deri, A., Tevan, G., Semlyen, A. and Castanheira, A., The Comlex Ground Return Plane, a Simlified Model for Homogeneous and Multi-layer Earth Return, IEEE Trans. PAS, vol., no. 8, , 98. [8] Tavares, M.C., Pissolato, J. and Portela, C., New Multihase Mode Domain Transmission Line Model - International Journal of Electrical Power & Energy Systems, Vol. 2/8, 585-6, Set [9] Tavares, M.C., Pissolato, J. and Portela, C., Mode Domain Multihase Transmission Line - Use in Transient Studies - IEEE Transactions on Power Delivery, vol. 4, No. 4, , Oct [] Tavares, M.C., Pissolato, J. and Portela, C., - Quasi- Modes Three-Phase Transmission Line Model - Comaring with Existing Frequency Deendent Models - Electric Power Systems Research International Journal, vol. 56, No. 2, , Nov. 2.