Holomorphic Embedding Load-flow method: Difference between revisions

Revision as of 12:23, 30 July 2013

Revision as of 20:50, 15 October 2015

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<ref name="FDLF">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.</ref>

, 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 a well-known fact 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_methods method#Complex_functionsComplex 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.

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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 intensiti es 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 [[Groebner_basisGroebner basis#Elimination_propertyElimination property|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.

The technique to obtain the coefficients for the power series expansion (on {{math|<var>s</var>{{=}}0}}) of voltages {{math|<var>V</var>}} is quite straightforward, once one realizes that equations ({{EquationNote|2}}) can be used to obtain them order after order. Consider the power series expansion for <math>\textstyle V(s)=\sum_{n = 0}^\infty a[n] s^n</math> and <math>\textstyle 1/V(s)=\sum_{n = 0}^\infty b[n] s^n</math>. By substitution into equations ({{EquationNote|1}}) and identifying terms at each order in {{math|<var>sⁿ</var>}}, one obtains:

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Once the power series at {{math|<var>s</var>{{=}}0}} are calculated to the desired order, the problem of calculating them at {{math|<var>s</var>{{=}}1}} becomes one of [[analytic continuation]]. It should be strongly remarked that this does not have anything in common with the techniques of [[Homotopy#Applications|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).

It can be proven<ref>L. Ahlfors, ''Complex analysis (3rd ed.)'', McGraw Hill, 1979.</ref> that algebraic curves are complete [[global analytic function]]s, 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_pointBranch point#Branch_cutsBranch cuts|branch cuts]]. Stahl’s extremal ___domain theorem<ref>G. A. Baker Jr and P. Graves-Morris, ''Padé Approximants'' (Encyclopedia of Mathematics and its Applications), Cambridge University Press, Second Ed. 2010, p. 326.</ref> 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 [[Conformal_radiusConformal radius#Version_from_infinityVersion from infinity:_transfinite_diameter_and_logarithmic_capacity transfinite diameter and logarithmic capacity|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<ref>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.</ref> 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_pointBranch point#Branch_cutsBranch cuts|branch cuts]] having minimal capacity.

These properties confer the load-flow method with the ability to unequivocally detect the condition of voltage collapse: the algebraic approximations are guaranteed to either converge to the solution if it exists, or not converge if the solution does not exist.