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<ref name=undefined /> {{Userspace draft|source=ArticleWizard|date=December 2011}} <!-- Please leave this line alone! -->
The '''Holomorphic Embedded Load Flow Method'''(HELM)
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This makes HELM particularly suited for real-time applications, and mandatory for any EMS software based on exploratory algorithms, such as contingency analysis, and under alert and emergency conditions solving operational limits violations and restoration providing guidance through action plans.<br />
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Consider the following general form for the load-flow equations:<br />
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A more detailed discussion about this procedure is included in the Holomorphic Embedding Wikipedia page, and a complete treatment is offered in Ref.<br/>
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Once the power series at <math>s=0</math> are calculated to the desired order, the problem of calculating them at <math>s=1</math> becomes one of [[analytic continuation]]. It should be strongly remarked that this does not have anything in common with the techniques of homotopic continuation. Homotopy is powerful since it only makes use of the concept of continuity and thus it is applicable to general smooth nonlinear systems, but on the other hand it does not always provide a reliable method to approximate the functions (as it relies on iterative schemes such as Newton-Raphson).<br/><br/>
It can be proven that algebraic curves are complete global analytic functions, that is, knowledge of the power series expansion at one point (the so-called germ of the function) uniquely determines the function everywhere on the complex plane, except on a finite number of branch cuts. Stahl’s extremal ___domain theorem further asserts that there exists a maximal ___domain for the analytic continuation of the function, which corresponds to the choice of branch cuts with minimal logarithmic capacity measure. In the case of algebraic curves the number of cuts is finite, therefore it would be feasible to find maximal continuations by finding the combination of cuts with minimal capacity. For further improvements, Stahl’s theorem on the convergence of Padé Approximants states that the diagonal and supra-diagonal Padé (or equivalently, the continued fraction approximants to the power series) converge to the maximal analytic continuation. The zeros and poles of the approximants remarkably accumulate on the set of branch cuts having minimal capacity.<br/><br/>
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== References ==
{{Reflist}}
<li>E. L. Allgower and K. Georg, "Introduction to Numerical Continuation Methods", SIAM Classics in Applied Mathematics 45, 2003.</li>
<li>This is a general phenomenon affecting the Newton-Raphson method when applied to equations in complex variables. See for instance Newton's_method#Complex_functions.</li>
<li>US patent 7519506, Antonio Trias, "System and method for monitoring and managing electrical power transmission and distribution networks", issued 2009-04-14; US patent 7979239, Antonio Trias, "System and method for monitoring and managing electrical power transmission and distribution networks", issued 2011-07-12, assigned to Aplicaciones en Informatica Avanzada, S.A.</li>
<li>a b 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.</li>
<li>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.</li>
<li>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.</li>
<li>It is a well-known fact 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.</li>
<li>Newton's_method#Complex_functions</li>
<li>Problems with Iterative Load Flow, Elequant, 2010.</li>
<li>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.</li>
<li>B. Sturmfels, "Solving Systems of Polynomial Equations”, CBMS Regional Conference Series in Mathematics 97, AMS, 2002.</li>
<li>A. Trias and J. L. Marin, "Holomorphic Embedding Loadflow", IEEE Peprint, 2011.
Toni It has already been done the provision of a preprint of a future publication. Do you want JL to coauthor, only him, someonelse?
L. Ahlfors, Complex analysis (3rd ed.), McGraw Hill, 1979.</li>
<li>G. A. Baker Jr and P. Graves-Morris, Padé Approximants (Encyclopedia of Mathematics and its Applications), Cambridge University Press, Second Ed. 2010, p. 326.</li>
<li>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.</li>
<li>SuiteSparse, T. Davis, U. de Florida.</li>
== External links ==
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