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m power-flow w/hyphen and I suppose load-flow too. ndash. |
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The '''Holomorphic Embedding Load-flow Method''' ('''HELM'''){{nnbsp}}<ref group="note">HELM is a trademark of Gridquant Inc.</ref> is a solution method for the [[
The HELM load
{{cite patent
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== Background ==
The [[
{{NumBlk|:|<math>\sum_k Y_{ik} V_k + Y_i^{\text{sh}} V_i = \frac{S_i^*}{V_i^*}</math>|{{EquationRef|1}}}}
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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]],<ref>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.
* 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.
* 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.
▲<ref>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.</ref>
, which has poor convergence properties but very little memory requirements and is
straightforward to implement; the full [[Newton–Raphson method]]<ref>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.
''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.
* J. S. Thorp and S. A. Naqavi, "Load flow fractals", ''in Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 2, pp.
▲, 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<sup><var>N</var></sup>}} 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.
* 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.
* 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.
</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.
▲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.
The key differential advantage of the HELM is that it is fully deterministic and unambiguous: it guarantees that the solution always
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{{NumBlk|:|<math>\sum_k Y_{ik} V_k(s) + Y_i^{\text{sh}} V_i(s) = s\frac{S_i^*}{V_i ^*(s^*)}</math>|{{EquationRef|1}}}}
With this choice, at {{math|<var>s</var>{{=}}0}} the right hand side terms become zero, (provided that the denominator is not zero), this corresponds to the case where all the injections are zero and this case has a well known and simple operational solution: all voltages are equal and all flow intensities are zero. Therefore, this choice for the embedding provides at s=0 a well known operational solution.
Now using classical techniques for variable elimination in polynomial systems<ref>B. Sturmfels, "Solving Systems of Polynomial Equations”, CBMS Regional Conference Series in Mathematics 97, AMS, 2002.</ref> (results from the theory of [[Resultants]] and [[Gröbner basis#Elimination|Gröbner basis]] it can be proven that equations ({{EquationNote|1}}) do in fact define {{math|<var>V(s)</var>}} as holomorphic functions. More significantly, they define {{math|<var>V(s)</var>}} as [[algebraic curves]]. It is this specific fact, which becomes true because the embedding is holomorphic that guarantees the uniqueness of the result. The solution at {{math|<var>s</var>{{=}}0}} determines uniquely the solution everywhere (except on a finite number of branch cuts), thus getting rid of the multi-valuedness of the load-flow problem.
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== See also ==
▲* [[Power flow study]]
* [[Power system simulation]]
* [[Unit commitment problem in electrical power production]]
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