Holomorphic Embedding Load-flow method: Difference between revisions

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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,<ref group="note" name="multsol">It is well-known that the load-flow equations for a power system have multiple solutions. For a network with {{math|<var>N</var>}} non-swing buses, the system may have up to {{math|2^<var>N</var>}} 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.</ref> 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.<ref group="note">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]].</ref> 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.<ref>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.

</ref> A simple illustration for the two-bus model is provided in<ref>[http://www.elequant.com/products/agora/demo/iterativeloadflow/ Problems with Iterative Load Flow] {{Webarchive|url=https://web.archive.org/web/20100104180641/http://www.elequant.com/products/agora/demo/iterativeloadflow/ |date=2010-01-04 }}, Elequant, 2010.</ref> Although there exist [[Homotopy|homotopic]] [[Numerical continuation|continuation]] techniques that alleviate the problem to some degree,<ref>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.</ref> the fractal nature of the basins of attraction precludes a 100% reliable method for all electrical scenarios.