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HandWiki. Holomorphic Embedding Load Flow Method. Encyclopedia. Available online: https://encyclopedia.pub/entry/36547 (accessed on 03 October 2026).
HandWiki. Holomorphic Embedding Load Flow Method. Encyclopedia. Available at: https://encyclopedia.pub/entry/36547. Accessed October 03, 2026.
HandWiki. "Holomorphic Embedding Load Flow Method" Encyclopedia, https://encyclopedia.pub/entry/36547 (accessed October 03, 2026).
HandWiki. (2022, November 25). Holomorphic Embedding Load Flow Method. In Encyclopedia. https://encyclopedia.pub/entry/36547
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Holomorphic Embedding Load Flow Method
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The content is sourced from: https://handwiki.org/wiki/Engineering:Holomorphic_embedding_load_flow_method

The Holomorphic Embedding Load-flow Method (HELM) is a solution method for the power flow equations of electrical power systems. Its main features are that it is direct (that is, non-iterative) and that it mathematically guarantees a consistent selection of the correct operative branch of the multivalued problem, also signalling the condition of voltage collapse when there is no solution. These properties are relevant not only for the reliability of existing off-line and real-time applications, but also because they enable new types of analytical tools that would be impossible to build with existing iterative load flow methods (due to their convergence problems). An example of this would be decision-support tools providing validated action plans in real time. The HELM load flow algorithm was invented by Antonio Trias and has been granted two US Patents. A detailed description was presented at the 2012 IEEE PES General Meeting and subsequently published. The method is founded on advanced concepts and results from complex analysis, such as holomorphicity, the theory of algebraic curves, and analytic continuation. However, the numerical implementation is rather straightforward as it uses standard linear algebra and the Padé approximation. Additionally, since the limiting part of the computation is the factorization of the admittance matrix and this is done only once, its performance is competitive with established fast-decoupled loadflows. The method is currently implemented into industrial-strength real-time and off-line packaged EMS applications.

linear algebra numerical implementation holomorphicity

References

  1. J. B. Ward and H. W. Hale, "Digital Computer Solution of Power-Flow Problems," Power Apparatus and Systems, Part III. Transactions of the American Institute of Electrical Engineers, vol.75, no.3, pp.398-404, Jan. 1956. A. F. Glimn and G. W. Stagg, "Automatic Calculation of Load Flows", Power Apparatus and Systems, Part III. Transactions of the American Institute of Electrical Engineers, vol.76, no.3, pp.817-825, April 1957. Hale, H. W.; Goodrich, R. W.; , "Digital Computation or Power Flow - Some New Aspects," Power Apparatus and Systems, Part III. Transactions of the American Institute of Electrical Engineers, vol.78, no.3, pp.919-923, April 1959.
  2. W. F. Tinney and C. E. Hart, "Power Flow Solution by Newton's Method," IEEE Transactions on Power Apparatus and Systems, vol. PAS-86, no.11, pp.1449-1460, Nov. 1967. S. T. Despotovic, B. S. Babic, and V. P. Mastilovic, "A Rapid and Reliable Method for Solving Load Flow Problems," IEEE Transactions on Power Apparatus and Systems, vol. PAS-90, no.1, pp.123-130, Jan. 1971.
  3. B. Stott and O. Alsac, "Fast Decoupled Load Flow," IEEE Transactions on Power Apparatus and Systems, vol. PAS-93, no.3, pp.859-869, May 1974.
  4. It is well-known that the load flow equations for a power system have multiple solutions. For a network with N non-swing buses, the system may have up to 2N possible solutions, but only one is actually possible in the real electrical system. This fact is used in stability studies, see for instance: Y. Tamura, H. Mori, and S. Iwamoto,"Relationship Between Voltage Instability and Multiple Load Flow Solutions in Electric Power Systems", IEEE Transactions on Power Apparatus and Systems, vol. PAS-102 , no.5, pp.1115-1125, 1983.
  5. This is a general phenomenon affecting the Newton-Raphson method when applied to equations in complex variables. See for instance Newton's method.
  6. R. Klump and T. Overbye, “A new method for finding low-voltage power flow solutions", in IEEE 2000 Power Engineering Society Summer Meeting,, Vol. 1, pp. 593-–597, 2000. J. S. Thorp and S. A. Naqavi, "Load flow fractals", in Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 2, pp. 1822--1827, 1989. J. S. Thorp, S. A. Naqavi, and H. D. Chiang, "More load flow fractals", in Proceedings of the 29th IEEE Conference on Decision and Control, Vol. 6, pp. 3028--3030, 1990. S. A. Naqavi, Fractals in power system load flows, Cornell University, August 1994. J. S. Thorp, and S. A. Naqavi, S.A., "Load-flow fractals draw clues to erratic behaviour", IEEE Computer Applications in Power, Vol. 10, No. 1, pp. 59--62, 1997. H. Mori, "Chaotic behavior of the Newton-Raphson method with the optimal multiplier for ill-conditioned power systems", in The 2000 IEEE International Symposium on Circuits and Systems (ISCAS 2000 Geneva), Vol. 4, pp. 237--240, 2000.
  7. Problems with Iterative Load Flow , Elequant, 2010. http://www.elequant.com/products/agora/demo/iterativeloadflow/
  8. V. Ajjarapu and C. Christy, "The continuation power flow: A tool for steady state voltage stability analysis", IEEE Trans. on Power Systems, vol.7, no.1, pp. 416-423, Feb 1992.
  9. B. Sturmfels, "Solving Systems of Polynomial Equations”, CBMS Regional Conference Series in Mathematics 97, AMS, 2002.
  10. A. Trias, "The Holomorphic Embedding Load Flow Method", IEEE Power and Energy Society General Meeting 2011, 22–26 July 2012.
  11. L. Ahlfors, Complex analysis (3rd ed.), McGraw Hill, 1979.
  12. G. A. Baker Jr and P. Graves-Morris, Padé Approximants (Encyclopedia of Mathematics and its Applications), Cambridge University Press, Second Ed. 2010, p. 326.
  13. H. Stahl, “The Convergence of Padé Approximants to Functions with Branch Points”, J. Approx. Theory, 91 (1997), 139-204. G. A. Baker Jr and P. Graves-Morris, Padé Approximants (Encyclopedia of Mathematics and its Applications), Cambridge University Press, Second Ed. 2010, p. 326-330.
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