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Modeling lightning strike behavior in the near field of elevated systems

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Abstract

The modeling of lightning strike behavior and estimation of the subsequent electric discharge is of great practical importance. In this study, a complete two-dimensional physics-based analytic formulation is presented for elevated grounded systems that can be envisioned to be contained within two non-concentric circular domains. The inner circle encompasses the body or system of interest, and the periphery of the outer circle addresses the cloud coverage. The potential field between the circular domains is modeled as the sum of two separate contributions. The first is formulated in terms of an eigen-function expansion involving simple radial functions and Legendre polynomials, while the second contribution is developed using two different approaches. The first approach utilizes an eigen-function expansion incorporating spherical Bessel functions and Legendre polynomials, while the second approach uses a Green’s function expansion also involving orthogonal polynomial functions. Each of the contributions to the total potential field leads to linear systems of equations that are solved for the unknown series expansion coefficients. The accuracy of the potential field solution is investigated with regard to convergence, stability, and error compared with an exact solution. The potential field solution is then used as the basis to evaluate leader formation, as a function of elevation, extent of cloud coverage, and propagation angle of the downward leader. Regions of high risk to lightning strikes on the grounded structure are developed in terms of joint probability functions. The electric discharge is estimated using electric current and is shown to be in the range of presently available information.

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References

  1. Schonland BJ (1964) The flight of thunderbolts. Clarendon Press, Oxford

    Google Scholar 

  2. Uman MA (1969) Lightning. McGraw Hill, New York

    Google Scholar 

  3. Uman MA, Rakov VA (2003) The interaction of lightning with airborne vehicles. Prog Aerosp Sci 39:61–81

    Article  Google Scholar 

  4. Miyake K, Suzuki T, Takashima M, Takuma M, Tada T (1990) Winter lightning on Japan sea coast: lightning striking frequency to tall structures. IEEE Trans Power Deliv 5:1370–1376

    Article  Google Scholar 

  5. Sorensen T, Brask MH, Grabau P, Olsen K (1998) Lightning damages to power generation wind turbines. In: Proceedings of the 24th international conference on lightning protection

  6. Lu G, Winn WP, Sonnenfeld RG (2011) Charge transfer during intracloud lightning from a time dependent multi-dipole model. J Geophys Res 116:1–16

    Google Scholar 

  7. Lu G, Cummer SA, Blakeslee RJ, Weiss S, Beasley WH (2012) Lightning morphology and impulse charge moment change of high peak current negative strokes. J Geophys Res 117:1–16

    Google Scholar 

  8. Rakov VA, Uman MA (2003) Lightning physics and effects. Cambridge University Press, Cambridge

    Book  Google Scholar 

  9. Krider EP (1986) Physics of lightning, studies in geophysics: the earth’s electrical environment. National Academy Press, Washington

    Google Scholar 

  10. Orville RE (1968) A high speed time-resolved spectroscopic study of lightning return stroke. J Atmos Sci 25:827–856

    Article  ADS  Google Scholar 

  11. Krider EP, Guo C (1983) The peak electromagnetic power radiated by lightning return strokes. J Geophys Res 88:8471–8474

    Article  ADS  Google Scholar 

  12. Berger K, Anderson RB, Kroeninger H (1975) Parameters of lightning flashes. Electra 41:23–37

    Google Scholar 

  13. Anderson RB, Eriksson AJ (1980) Lightning parameters for engineering application. Electra 69:65–102

    Google Scholar 

  14. Miyake K, Suzuki T, Shinjou K (1992) Characteristics of winter lighting current on Japan Sea Coast. IEEE Trans Power Deliv 7:1450–1457

    Article  Google Scholar 

  15. Saba MF, Schultz W, Warner TA, Campos LZ, Schumann C, Krider EP, Cummins KL, Orville RE (2010) High-speed video observations of positive lightning flashes to ground. J Geophys Res 115:1–9

    Article  Google Scholar 

  16. International Electrotechnical Commission IEC (2010) IEC 61400–24 standard, wind turbines—part 24: lightning protection. IEC Press, Geneva

  17. International Electrotechnical Commission IEC (2010) International Electrotechnical Commission, IEC 62305–1 standard, protection against lightning—part 1: general principles. IEC Press, Geneva

  18. Berger K (1967) Novel observations on lightning discharges: results of research on Mount San Salvatore. J Frankl Inst 283:478–525

    Article  Google Scholar 

  19. Eriksson AJ (1978) Lightning and tall structures. Trans S Afr Inst Electr Eng 69:238–252

    Google Scholar 

  20. Golde RH (1978) Lightning and tall structures. IEEE Proc 125:347–351

    Google Scholar 

  21. Rizk FM (1994) Modeling of lightning incidence to tall structures. Part I: theory. Trans Power Deliv 9:162–170

    Article  Google Scholar 

  22. Bondiou A, Gallimberti I (1994) Theoretical modeling of the development of the positive spark in long gaps. J Phys D 27:1252–1266

    Article  ADS  Google Scholar 

  23. Dellera L, Garbagnati E (1990) Lightning stroke simulation by means of the leader progression model. Part I, description of the model and evaluation of exposure of free standing structure. IEEE Trans Power Deliv 5:2009–2017

    Article  Google Scholar 

  24. Goelian N, Lalande P, Bondiou-Clergerie A, Bacchiega GL, Gazzani A, Gallimberti I (1997) A simplified model for the simulation of positive-spark development in long air gaps. J Phys D 30:2441–2452

    Article  ADS  Google Scholar 

  25. Szedenik N (2001) Rolling sphere. J Electrost 51:345–350

    Article  Google Scholar 

  26. Becerra M, Cooray V (2006) A simplified physical model to determine the lightning upward connecting leader inception. IEEE Trans Power Deliv 21:897–908

    Article  Google Scholar 

  27. Becerra M, Cooray V, Hartono ZA (2007) Identification of lightning vulnerability points on complex grounded structures. J Electrost 65:562–570

    Article  Google Scholar 

  28. Becerra M, Cooray V, Roman F (2008) Lightning striking distance of complex structures. IET Gen Transm Distrib 2:131–138

    Article  Google Scholar 

  29. Bertelsen K, Erichsen HV, Jensen MS, Madsen SF (2007) Application of numerical models to determine lightning attachment points on wind turbines. In: Proceedings of the 30th international conference on lightning and static electricity

  30. Cooray V, Becerra M (2012) Attractive radii of vertical and horizontal conductors evaluated using a self-consistent leader inception and propagation model-SLIM. J Atmos Res 117:64–70

    Article  Google Scholar 

  31. Madsen SF, Erichsen HV (2009) Numerical model to determine lightning attachment point distributions on wind turbines according to the revised IEC 61400–24. In: Proceedings of the international conference on lightning and static electricity, Pittsfield, Massachusetts

  32. Madsen SF, Bertelsen K, Krogh TH, Erichsen HV, Hansen AN, Lonbaek KB (2012) Proposal of new zoning concept considering lightning protection of wind turbine blades. J Light Res 4:108–117

    Article  Google Scholar 

  33. Gao Y, Lu W, Ma Y, Chen L, Zhang Y, Yan X, Zhang Y (2014) Three dimensional propagation characteristics of the upward connecting leaders in six negative tall object flashes in Guangzhou. J Atmos Res 149:193–203

    Article  Google Scholar 

  34. Asmar NH (2005) Partial differential equations with Fourier series and boundary value problems. Prentice Hall, Upper Saddle River

    Google Scholar 

  35. Cooray V, Rakov V, Theethayi N (2007) The lightning strike distance-revisited. J Electrost 65:296–306

    Article  Google Scholar 

  36. Morse PM, Feshbach H (1953) Methods of theoretical physics, vol 2. McGraw Hill Company, New York

    MATH  Google Scholar 

  37. Borse GJ (1997) Numerical methods with Matlab: a resource for scientists and engineers. PWS Publishing Company, Boston

    Google Scholar 

  38. Arfken GB, Weber HJ (2005) Mathematical methods for physicists. Academic Press, Charlottesville

    MATH  Google Scholar 

  39. Jackson JD (1999) Classical electrodynamics. Wiley, Hoboken

    MATH  Google Scholar 

  40. MacRobert TM (1967) Spherical harmonics; an elementary treatise on harmonic functions with applications. Pergamon Press, New York

    MATH  Google Scholar 

  41. Armstrong HR, Whitehead ER (1968) Field and analytical studies of transmission line shielding. IEEE Trans Power Appar Syst 87:270–281

    Article  Google Scholar 

  42. Roberts J, Roberts TD (1978) Use of the butterworth low-pass filter for oceanographic data. J Geophys Res Oceans 83:5510–5514

    Article  Google Scholar 

  43. Trefethen LN (1992) Pseudospectra of matrices. In: Griffiths DF, Watson GA (eds) Proceedings of the 14th Dundee biennial conference on numerical analysis. Longman Scientific and Technical, Essex

  44. Trefethen LN, Embree M (2005) Spectra and pseudospectra: the behavior of non-normal matrices and operators. Princeton University Press, Princeton

    MATH  Google Scholar 

  45. Wright TG (2002) Eigtool: a graphical tool for non-symmetric eigenvalue problems. http://web.comlab.ox.ac.uk/projects/pseudospectra/software.html. Accessed 15 June 2013

  46. Hewitt E, Hewitt RE (1979) The Gibbs-Wilbraham phenomenon: an episode in Fourier analysis. Arch Hist Exact Sci 21:129–160

    Article  MathSciNet  MATH  Google Scholar 

  47. Lai M, Lin W, Wang W (2002) A fast spectral-difference method without pole conditions for Poisson-type equations in cylindrical and spherical geometries. J Numer Anal 22:537–548

    Article  MathSciNet  MATH  Google Scholar 

  48. Lai M, Tseng J (2007) A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical and spherical geometries. White paper, National Chiao Tung University, Taiwan. http://jupiter.math.nctu.edu.tw. Accessed 15 Feb 2014

  49. Reese OW (1971) National Aeronautics and Space Administration, technical note, NASA TN D-6438: numerical method and Fortran program for solving Poisson’s equation over axisymmetric regions of a sphere. NASA, Langley

  50. Roberts B, Shepard D, Caldeira K, Cannon M, Eccles D, Grenier A (2007) Harnessing high-altitude wind power. IEEE Trans Energy Convers 22:136–144

    Article  Google Scholar 

Download references

Acknowledgments

The authors gratefully acknowledge the partial financial support of the Wofford Cain ’13 Senior Endowed Chair in Offshore Technology at Texas A&M University.

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Authors and Affiliations

  1. Ocean Engineering Program, Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, College Station, TX, 77843-3136, USA

    G. A. Malinga

  2. Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, College Station, TX, 77843-3136, USA

    J. M. Niedzwecki

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  1. G. A. Malinga
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Correspondence to J. M. Niedzwecki.

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Malinga, G.A., Niedzwecki, J.M. Modeling lightning strike behavior in the near field of elevated systems. J Eng Math 97, 195–221 (2016). https://doi.org/10.1007/s10665-015-9805-y

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  • DOI: https://doi.org/10.1007/s10665-015-9805-y

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