Lightning Transients on Branched Distribution Lines Considering Frequency-Dependent Ground Parameters

Transcription

1 214 International Conference on Lihtnin Protection (ICLP), Shanhai, China Lihtnin Transients on Branched Distribution Lines Considerin Frequency-Dependent Ground Parameters Alberto De Conti LRC Lihtnin Research Center UFMG Federal University of Minas Gerais Belo Horizonte, Brazil Rafael Alipio DEE Department of Electrical Enineerin CEFET-MG Federal Center of Technoloical Education Belo Horizonte, Brazil Abstract This paper investiates the influence of considerin frequency-dependent round parameters in the calculation of lihtnin transients on a two-phase distribution line includin the presence of branches. The analysis is focused on propaation effects alon the line. It is shown that the influence of frequencydependent round parameters is only remarkable if a poorlyconductin round is considered. It is also shown that the presence of line branches is likely to reduce the relative importance of considerin the frequency variation of the round parameters in the calculation of lihtnin transients. Keywords- transmission line modelin; frequency-dependent round parameters; branched distribution lines; lihtnin overvoltaes. I. INTRODUCTION There has been an increasin interest in the simulation of electromanetic transients considerin the variation of the round conductivity and permittivity with frequency [1-14]. However, most of the available literature deals with line topoloies in which the presence of branches and multiple roundin points is nelected. More recently, an attempt was made to investiate to what extent the consideration of frequency-dependent round parameters would be important in the calculation of switchin transients on a branched power distribution line [15]. In this paper, this analysis is extended to the study of direct lihtnin strikes over a power distribution line with complex topoloy, with focus on the influence of incorporatin frequency-dependent round parameters on the transmission line model. II. MODELING ASSUMPTIONS A. Transmission Line Parameters The transmission line parameters are calculated assumin the round-return impedance to be iven by 2hi j e Z ii d 2 2 ( h ) i h j j e Z cos( ) ij rij d 2 2 where j j( k) ] [ r In (1)-(3), is the anular frequency, is the round conductivity, r is the round relative permittivity, h i and h j are the heihts of conductors i and j, r ij is the horizontal separation between the conductors, =41 7 H/m, = F/m, and k is a constant that selects the desired round-return model [15,16]. If k=1 in (3), equations (1) and (2) reduce to Nakaawa s equations [17]; if k= r, equations (1) and (2) reduce to Carson s equations [18], in which it is implicitly assumed that r is equal to the relative permittivity of the vacuum [16]. The fact that it is impossible to set values of round relative permittivity other than unity in Carson s equations limits their application to cases in which >>. In Nakaawa s equations it is possible to set any desired value to r, which makes them applicable to studyin hih-frequency transients in transmission lines located above hih-resistivity soils. B. Soil Model The considered soil model is the one proposed in [4]. It is based on measurements of the frequency response of 65 different types of soils in Brazil. This soil model describes the variation of the round parameters with frequency as This work was supported by The State of Minas Gerais Research Foundation (FAPEMIG), under rants TEC-APQ and TEC-PPM , and by the National Council of Scientific and Technoloical Development (CNPq), under rant 36195/213-5.

2 f ( ) h( ) (3) tan( n / 2) 1 n1 r ( ) 6 h( ) f (4) n 2 1 where () is the frequency-dependent soil conductivity, in ms/m, is the low-frequency soil conductivity determined at 1 Hz, in ms/m, r () is the frequency-dependent relative permittivity, / =12 is the relative permittivity at hiher.73 frequencies, f is the frequency, h ( ) 1. 26, and n=.54 (see [4] for details). C. Transmission Line Model All analyses consider the modal-domain transmission line model of Marti [19] extended to include frequency-dependent round parameters as outlined in [15]. The transmission line parameters are calculated in MATLAB. The modal propaation functions and the modal characteristic impedances are fitted in the frequency domain from 1 Hz to 1 MHz usin the vector fittin technique [2]. The resultin poles and residues are written in the form of a.pch file that is read and interpreted by the Alternative Transients Proram (ATP) as a frequencydependent line. Details of the implemented procedure can be found in [15]. III. SIMULATED CASES Fi. 1 illustrates the three basic cases considered in the analysis. The solid lines represent two-phase power distribution lines that are typically used in rural areas in Brazil. A vertically-stacked confiuration is assumed in which the top and bottom conductors are at heihts of 8.4 m and 7.2 m above the round. The conductors are represented as solid copper wires of 5 mm radius, whose internal impedance is calculated with the traditional formulation usin Bessel s functions. In practical conditions, the bottom conductor is expected to be rounded every 2 m or so. Here, it is not rounded to keep the analysis focused on the effect of frequency-dependent round parameters on the wave propaation alon the transmission lines. In all cases, all line terminations were left open, except at point A where both conductors were connected to round by means of resistances of 48. This value approaches the characteristic impedances of both conductors if losses are nelected. Case 1 assumes a transmission line with lenth of 12 m without any branches. Cases 2 and 3 add laterals with lenths L n to Case 1, where n corresponds to the lateral number. All transmission line sements were simulated via.pch files in ATP as outlined in [15]. In all analyses, an ideal current source with a normalized peak amplitude of 1 A was considered. A lihtnin current waveform represented as a sinle Heidler function with virtual front time t d3 =.2 s (measured as the time from.3i p to.9i p divided by.6, where I p is the current peak value) and time to half-value of 21 s was assumed. This current waveform can n be considered representative of fast subsequent stroke currents measured at short instrumented towers [21, 22]. This lihtnin current was injected by an ideal current source at points A or C indicated in Fi. 1. IV. RESULTS AND ANALYSIS A. Case 1 Fi. 2 illustrates voltaes at points C and E for the injection of the lihtnin current at point A, considerin a low-frequency round conductivity of.1 S/m. The illustrated waveforms were obtained for the calculation of the transmission line parameters either with Carson s formula assumin a constant round conductivity (curves labeled, Carson ) or with Nakaawa s equations assumin frequency-dependent round parameters (curves labeled (), r (), Nakaawa ). Fi. 3 repeats the analysis of Fi. 2 assumin a low-frequency round conductivity of.1 S/m. The.pch file associated with the 15-m lon line sement considered as a buildin block for simulatin the cases referrin to frequency-dependent round parameters with =.1 S/m can be found in the appendix of [15]. 3 m Case 1 F G H L 1 L 2 L 3 Case 2 F G H L 1 L 2 L 3 L 4 L 5 L Case 3 Fiure 1. Investiated line topoloies. (), r (),Nakaawa, Carson Fiure 2. Voltaes at points C and E for Case 1 [line illustrated in Fi. 1] considerin the current injection at point A and a low-frequency round conductivity of.1 S/m.

3 (), r (),Nakaawa, Carson Fiure 3. Same as Fi. 2, but for a low-frequency round conductivity of.1 S/m. It is seen in Fi. 2 that for a round conductivity of.1 S/m the differences between the calculated voltae waveforms are not sinificant. Althouh not shown, for a more conductive round the observed differences become neliible. In Fi. 3, more noticeable differences are observed in the curves calculated considerin or nelectin the variation of the round parameters with frequency for a round conductivity of.1 S/m. Althouh the calculated peak values are not sinificantly affected, the observed waveform distortion could be determinant for the occurrence or not of insulation breakdown in certain conditions. Similar conclusions apply if the lihtnin current is injected at point C, as seen in Fi. 4, which shows the voltaes calculated at points B, C, D and E for a low-frequency round conductivity of.1 S/m. B. Case 2 The analysis presented in this section considers Case 2, which corresponds to the distribution line with three laterals illustrated in Fi. 1. In the simulations, the laterals were assumed to have the followin lenths: L 1 =15 m, L 2 =3 m, and L 3 =45 m. Fi. 5 illustrates voltaes calculated at points B, C, D, and E assumin the current injection at point A for a lowfrequency round conductivity of.1 S/m, considerin or nelectin the frequency variation of the round parameters as before. Fi. 6 illustrates voltaes calculated for the current injection at point C. It is seen in Fis. 5 and 6 that the inclusion of laterals modifies the calculated waveforms in comparison with the results presented in the previous section, which were obtained for a transmission line without any branches. This is the result of the multiple reflections that take place at the various openended terminations. Another effect associated with the inclusion of laterals is the reduction of the peak voltaes for both current injection points. A comparison of the voltae waveforms calculated considerin or nelectin the variation of the round parameters with frequency shows that includin this variation modifies the fine structure of the waveforms without chanin their overall characteristics. The most notable differences between the calculated waveforms are observed at point E for the injection of the lihtnin current at point C, as illustrated in Fi. 6. However, even in this case the calculated peak voltaes are not affected. 5 (), 4 r (),Nakaawa 3, Carson Fiure 4. Voltaes at points B, C, D, and E for Case 1 [line illustrated in Fi. 1] considerin the current injection at point C and a low-frequency round conductivity of.1 S/m (), r (),Nakaawa, Carson Fiure 5. Voltaes at points B, C, D, and E for Case 2 [line illustrated in Fi. 1] considerin the current injection at point A and a low-frequency round conductivity of.1 S/m.

4 (), r (),Nakaawa, Carson Fiure 6. Same as Fi. 5, but for current injection at point C. C. Case 3 The final set of results presented in this paper refers to Case 3, which corresponds to the distribution line with six laterals illustrated in Fi. 1. In the simulations, the followin lenths were assumed for the laterals: L 1 =L 4 =L 6 =15 m, L 2 =3 m, L 3 =45 m, and L 5 =6 m. Fi.7 illustrates the voltaes at points B, C, D, and E for the current injection at point A. Fi.8 does the same, but for the current injection at point C. In both cases, a low-frequency round conductivity of.1 S/m was assumed. It is seen in Fis. 7 and 8 that the consideration of a total of six laterals reduces even further the calculated peak values. For example, for the current injection at point A the peak voltae at point E reaches about 45 kv in Case 1, 39 kv in Case 2, and 34 kv in Case 3. For the current injection at point C, the peak voltae at point E reaches about 525 kv in Case 1, 443 kv in Case 2, and 375 kv in Case 3. Aain, as expected, this reduction is associated with the multiple reflections that take place in the branches. Interestinly, in the presence of six laterals the calculated voltae waveforms are nearly model independent reardless of the assumed current injection point. This suests that considerin the frequency variation of the round parameters miht not be important in the calculation of the transmission line parameters if an analysis of lihtnin overvoltaes on hihly-branched distribution lines is to be performed (), 1 r (),Nakaawa, Carson Fiure 7. Voltaes at points B, C, D, and E for Case 3 [line illustrated in Fi. 1] considerin currin injection at point A and a low-frequency round conductivity of.1 S/m (), r (),Nakaawa, Carson Fiure 8. Same as Fi. 7, but for current injection at point C.

5 D. Discussion The results presented in the previous sections indicate that the inclusion of line branches is likely to reduce the importance of considerin the frequency variation of round parameters in the calculation of transmission line parameters if lihtnin overvoltaes are to be calculated on distribution lines. In fact, dependin on the system topoloy the use of Carson s formula with constant round parameters can lead to acceptable results even for a value of low-frequency round conductivity as low as.1 S/m. It must be reminded that the considered lihtnin current has a virtual front time of.2 s, which approaches the lower limit of front times of subsequent return-stroke currents measured at short instrumented towers. If slower lihtnin currents were considered, the differences observed between the voltae waveforms calculated assumin or nelectin the variation of the round parameters with frequency would be even less sinificant. This happens because in this case the frequency content of the injected current would be more concentrated in the low-frequency rane, and the reduction in round resistivity and relative permittivity that is expected in the hih-frequency reion would be less relevant for the calculated results. Finally, most of the results presented in this section refer to a round conductivity of.1 S/m at lower frequencies. As suested in the results presented in Fi. 1, the consideration of a more conductive round would reduce even further the differences observed if constant or frequency-dependent round parameters are considered in the calculation of the transmission line parameters, especially in a line topoloy that include a reat number of laterals. V. CONCLUSIONS This paper investiates the influence of considerin a frequency-dependent soil in the assessment of lihtnin overvoltaes on a typical two-phase distribution line. The analysis is focused on propaation effects alon the line. It is shown that frequency-dependent round parameters are of some relevance only for very low values of round conductivity (e..,.1 S/m at low frequencies). It is also shown that the presence of multiple line branches is likely to reduce the relative importance of considerin a frequencydependent soil in the calculation of transmission line parameters even for a poorly-conductin round, at least in terms of resultin lihtnin transients. REFERENCES [1] C. Portela, M. C. Tavares, and J. Pissolato, Accurate Representation of Soil Behaviour for Transient Studies, IEE Proceedins on Generation, Transmission and Distribution, vol. 15, no. 6, pp , Nov. 23. [2] C. Portela, J. B. Gertrudes, M. C. Tavares, and J. Pissolato, Earth Conductivity and Permittivity Data Measurements Influence in Transmission Line Transient Performance, in Proceedins of IPST 25 International Conference on Power Systems Transients, Montreal, Canada, Jun. 25. [3] S. Visacro and R. 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