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The Holomorphic Embedding Load-flow Method (HELM〔HELM is a trademark of Gridquant Inc.〕) 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 flows (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 published in.〔A. Trias, "The Holomorphic Embedding Load Flow Method", ''IEEE Power and Energy Society General Meeting 2011'', 22–26 July 2012.〕 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 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. == Background == The load-flow calculation is one of the most fundamental components in the analysis of power systems and is the cornerstone for almost all other tools used in power system simulation and management. The load-flow equations can be written in the following general form: V_i = \frac|}} where the given (complex) parameters are the admittance matrix , the bus shunt admittances , and the bus power injections representing constant-power loads and generators. To solve this non-linear system of algebraic equations, traditional load-flow algorithms were developed based on three iterative techniques: the Gauss-Seidel method 〔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.〕 , which has poor convergence properties but very little memory requirements and is straightforward to implement; the full Newton-Raphson method 〔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.〕 , which has fast (quadratic) iterative convergence properties, but it is computationally costly; and the Fast Decoupled Load-Flow (FDLF) method 〔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.〕 , which is based on Newton-Raphson, but greatly reduces its computational cost by means of a decoupling approximation that is valid in most transmission networks. Many other incremental improvements exist; however, the underlying technique in all of them is still an iterative solver, either of Gauss-Seidel or of Newton type. There are two fundamental problems with all iterative schemes of this type. On the one hand, there is no guarantee that the iteration will always converge to a solution; on the other, since the system has multiple solutions,〔It is a well-known fact that the load flow equations for a power system have multiple solutions. For a network with non-swing buses, the system may have up to 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.〕 it is not possible to control which solution will be selected. As the power system approaches the point of voltage collapse, spurious solutions get closer to the correct one, and the iterative scheme may be easily attracted to one of them because of the phenomenon of Newton fractals: when the Newton method is applied to complex functions, the basins of attraction for the various solutions show fractal behavior.〔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.〕 As a result, no matter how close the chosen initial point of the iterations (seed) is to the correct solution, there is always some non-zero chance of straying off to a different solution. These fundamental problems of iterative loadflows have been extensively documented .〔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. 〕 A simple illustration for the two-bus model is provided in〔(Problems with Iterative Load Flow ), Elequant, 2010.〕 Although there exist homotopic continuation techniques that alleviate the problem to some degree,〔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.〕 the fractal nature of the basins of attraction precludes a 100% reliable method for all electrical scenarios. The key differential advantage of the HELM is that it is fully deterministic and unambiguous: it guarantees that the solution always corresponds to the correct operative solution, when it exists; and it signals the non-existence of the solution when the conditions are such that there is no solution (voltage collapse). Additionally, the method is competitive with the FDNR method in terms of computational cost. It brings a solid mathematical treatment of the load-flow problem that provides new insights not previously available with the iterative numerical methods. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Holomorphic embedding load flow method」の詳細全文を読む スポンサード リンク
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